diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzhkyp" "b/data_all_eng_slimpj/shuffled/split2/finalzzhkyp" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzhkyp" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{intro}\n\n\nIn his seminal article that anticipated the field of quantum information, Feynman argued that simulating quantum systems on classical computers takes an amount of time that scales exponentially with the size of the system, while the cost of quantum simulations can scale in polynomial time with system size \\cite{Feynman}. This possibility may offer a path forward for computational chemistry \\cite{Lloyd, Alan}. A quantum simulation algorithm for quantum chemical Hamiltonians enables the efficient calculation of properties such as energy spectra \\cite{Alan}, reaction rates \\cite{Dan, Ivan}, correlation functions \\cite{Ortiz}, and molecular properties \\cite{Ivan2} for molecules larger than those that are currently accessible through classical calculations.\n\nQuantum simulation of electronic structure requires a representation of fermions by systems of qubits. Significant progress has been made on efficient quantum simulation of fermions. In 1997, Abrams and Lloyd proposed a simulation scheme for fermions hopping on a lattice~\\cite{AL}. In 2002, Somma {\\it et al}. used the Jordan-Wigner to generalize the simulation scheme proposed by Abrams and Lloyd \\cite{Somma, JW}. The Jordan-Wigner transformation has since been used to outline a scalable quantum algorithm for the simulation of molecular electron dynamics, and to design an explicit quantum circuit for simulating a Trotter time-step of the molecular electronic Hamiltonian for H$_2$ in a minimal basis \\cite{Alan, James}. Further refinements of the Jordan-Wigner construction were made by Verstrate and Cirac \\cite{VC} and by Bravyi and Kitaev \\cite{BK}. From the point of view of fundamental physics, such constructions can be regarded as giving a negative answer to the question of whether fundamental fermi fields are required to explain observed fermionic degrees of freedom~\\cite{Ball}. Practically speaking, such constructions show that quantum computation of electronic structure does not suffer from an analog of the sign problem; that is, fermion antisymmetry represents no significant obstacle to efficient algorithms. \n\nTheoretical progress in quantum simulation has been accompanied by experimental successes. In 2010, Lanyon {\\it et al}.\\ calculated the energy spectrum of a hydrogen molecule using an optical quantum computer~\\cite{Lanyon}. For a review of photonic quantum simulators, see \\cite{AlanWalther}. Du {\\it et al}.\\ repeated this result to higher precision with NMR shortly thereafter \\cite{Du}. Digital quantum simulations of the kind considered in the present paper have been implemented in ion traps using up to 100 gates and 6 qubits~\\cite{Blatt2}. The progress of trapped ion quantum simulation is detailed in \\cite{Blatt}. \n\nQuantum computation of electronic structure has been the subject of simulation studies~\\cite{Alan,Veis1} and has been extended to cover relativistic systems~\\cite{Veis2}. The history of calculations in quantum chemistry provides a useful sequence of problems reaching from calculations that can be performed on experimental quantum computers today to calculations at the present research frontier~\\cite{Love}. Despite these promising results, the scaling of the number of gates required by the algorithm outlined in \\cite{Alan, James} remains challenging. It is a subject of active research to find improvements to the (polynomial) scaling of the cost of the algorithm described in~\\cite{Alan, James}. Several improvements are described in~\\cite{Jones}, and the techniques of that work could be combined with those of the present paper to further reduce the resource requirements. \n\n\\begin{figure}[ht]\n \\centerline{\\includegraphics[scale = .45]{figure1.png}}\n \\caption[Criterion for a successful simulation scheme]{ A simulation scheme first encodes fermionic states in qubits, then acts with the qubit operator representing the fermionic operator (obtained by the associated transformation), then inverts the encoding to obtain the resultant fermionic state. The criterion for a successful simulation scheme is that this procedure reproduces the action of the fermionic operator, i.e. that Path 1 is equivalent to Path 2, for all basis states --- in other words, that this diagram commutes.\\label{fig1}}\n\\end{figure}\n\nA fermionic simulation scheme can be broken into two pieces: first, to map occupation number basis vectors to states of qubits; and second, to represent the fermionic creation and annihilation operators in terms of operations on qubits in a way that preserves the fermionic anti-commutation relations, as illustrated in Figure~\\ref{fig1}. Previous simulation algorithms have used a straightforward mapping of fermionic occupation number basis states to qubit states that was originally defined by Zanardi in the context of entanglement \\cite{Zanardi,Somma,Alan}. The Jordan-Wigner transformation is then used to write the electronic Hamiltonian as a sum over products of Pauli spin operators acting on the qubits of the quantum computer. Subsequently the Hamiltonian terms $h_k$, where $\\hat{H} = \\sum_{k} h_k$, are converted into the unitary gates that are the corresponding time evolution operators. Even though the $h_k$ do not necessarily commute, their sequential execution on a quantum computer can be made to approximate the unitary propagator $e^{-i \\hat{H} t}$ through a Trotter decomposition \\cite{Trotter, Suzuki, QA,MikeIke}. Finally, the iterative phase estimation algorithm (IPEA) is used to approximate the eigenvalue of an input eigenstate \\cite{Alan, James, MikeIke}. \n\nIn this paper we treat the Trotterization process and IPEA as standard procedures. We develop the Bravyi-Kitaev basis and Bravyi-Kitaev transformation, both named after the authors who first proposed such a scheme \\cite{BK}, which provide a more efficient mapping between electronic Hamiltonians and qubit Hamiltonians. While the occupation number basis and the Jordan-Wigner transformation allow for the representation of a single fermionic creation or annihilation operator by $O(n)$ qubit operations, the Bravyi-Kitaev basis and transformation require only $O(\\log n)$ qubit operations to represent one fermionic operator. It is worth noting that Bravyi and Kitaev were concerned with exploring the power of fermions as the basic hardware units of a quantum computer, rather than with the simulation of fermions by qubits \\cite{BK}. However, understanding how the structure of fermionic systems can be employed to process information helps us understand how standard quantum information procedures can be used to simulate the structure of fermionic systems. We work out a detailed application of the Bravyi-Kitaev transformation to the operators that appear in quantum chemical Hamiltonians, providing a new way of mapping electronic Hamiltonians to qubit Hamiltonians. We also give explicit Pauli decompositions of the qubit operators derived from this new transformation for the quantum chemical Hamiltonian for H$_2$ in a minimal basis. We show that the quantum circuit for simulating a single first-order Trotter time-step of the Bravyi-Kitaev minimal basis molecular hydrogen Hamiltonian requires 30 single-qubit gates and 44 CNOT gates, as compared to 46 single-qubit gates and 36 CNOT gates for the Jordan-Wigner Hamiltonian derived in \\cite{James}. Finally, we show that a chemical-precision estimate of the ground state eigenvalue of the Bravyi-Kitaev Hamiltonian can be obtained in 3 first-order Trotter steps, with a total cost of 222 gates, while the Jordan-Wigner Hamiltonian requires 4 first-order Trotter steps for a total of 328 gates. Since the Bravyi-Kitaev transformation is known to be asymptotically more efficient, this result for the simplest possible case of molecular hydrogen in a minimal basis demonstrates the superior efficiency of the Bravyi-Kitaev method for all molecular quantum simulations.\n\nIn Section~\\ref{back} we will review basic quantum chemistry in second quantized form as well as the Jordan Wigner transformation. In Section~\\ref{alt} we discuss alternatives to the occupation number basis, including the Bravyi-Kitaev basis, which we go on to describe in detail in Section~\\ref{sets}. In Section~\\ref{bravkit} we present the Bravyi-Kitaev transformation, which allows us to represent creation and annihilation operators in the Bravyi-Kitaev basis. In Section~\\ref{pauli} we compute the products of these operators that occur in electronic structure Hamiltonians. In Section~\\ref{molham} we compute the molecular electronic structure Hamiltonian of H$_2$ in a minimal basis using the Bravyi-Kitaev basis and transformation. In Section~\\ref{trott} we make an explicit comparison between the Bravyi-Kitaev transformation and the Jordan Wigner transformation by simulating the Trotterization procedure. We close the paper with some conclusions about the utility of the Bravyi-Kitaev transformation.\n\n\\section{Background}\\label{back}\n\n\\subsection{Fermionic systems and second quantization}\n\nWe may describe fermionic systems using the formalism of second quantization, in which $n$ single-particle states can be either empty or occupied by a spinless fermionic particle. In the context of quantum chemistry these $n$ states represent spin orbitals, ideally one-electron energy eigenfunctions and often molecular orbitals found by the Hartree-Fock method~\\cite{Mcweeny,Szabo}. We consider a subspace of the full Fock space which is spanned by $2^n$ electronic basis states $\\ket{f_{n-1}\\ \\ldots\\ f_{0}}$, where $f_j \\in \\{0,1\\}$ is the occupation number of orbital $j$ (restricted to these values due to the Pauli exclusion principle). This is called the {\\em occupation number} basis.\n\nAny interaction of a fermionic system can be expressed in terms of products of the creation and annihilation operators $a^\\dag_j$ and $a_j$, for $j \\in \\{0,\\ldots,n\\!-\\!1\\}$. Due to the exchange anti-symmetry of fermions, the action of $a^\\dag_j$ or $a_j$ introduces a phase to the electronic basis state that depends on the occupancy of all orbitals with index less than $j$ in the occupation number representation. (One can choose instead to define these operators so that it is the occupation of orbitals with \nindex greater than $j$ that determines the phase --- the ordering of orbitals is arbitrary.) These operators act on occupation number basis vectors as follows:\n\n\\begin{eqnarray}\na^\\dag_j\\, \\ket{f_{n-1}\\ \\ldots\\ f_{j+1}\\ 0\\ f_{j-1}\\ \\ldots\\ f_{0}}\n& \\,=\\, \\bigl(-1\\bigr)^{\\sum_{s=0}^{j-1}f_s}\\,\n\\ket{f_{n-1}\\ \\ldots\\ f_{j+1}\\ 1\\ f_{j-1}\\ \\ldots\\ f_{0}}; \\\\\na^\\dag_j\\, \\ket{f_{n-1}\\ \\ldots\\ f_{j+1}\\ 1\\ f_{j-1}\\ \\ldots\\ f_{0}}\n& \\,=\\, 0;\\\\\na_j\\, \\ket{f_{n-1}\\ \\ldots\\ f_{j+1}\\ 1\\ f_{j-1}\\ \\ldots\\ f_{0}}\n& \\,=\\, \\bigl(-1\\bigr)^{\\sum_{s=0}^{j-1}f_s}\\,\n\\ket{f_{n-1}\\ \\ldots\\ f_{j+1}\\ 0\\ f_{j-1}\\ \\ldots\\ f_{0}}; \\\\\na_j\\, \\ket{f_{n-1}\\ \\ldots\\ f_{j+1}\\ 0\\ f_{j-1}\\ \\ldots\\ f_{0}}\n& \\,=\\, 0.\n\\end{eqnarray}\nThe canonical fermionic anti-commutation relations enforce the exchange anti-symmetry:\n\\begin{equation} [a_j,a_k]_+=0, \\qquad [a^\\dag_j,a^\\dag_k]_+=0, \\qquad [a_j,a^\\dag_k]_+=\\delta_{jk}\\mathbf{1}, \\end{equation}\nwhere the anti-commutator of operators $A$ and $B$ is defined by $[A,B]_+\\equiv AB+BA$.\n\nThe molecular electronic Hamiltonian of interest in the electronic structure problem is:\n\\begin{equation}\\label{molhameq}\n\\hat{H}=\\sum_{i,j}h_{ij}\\ a^\\dag_i a_j +\\frac{1}{2}\\sum_{i,j,k,l} h_{ijkl}\\ a^\\dag_i a^\\dag_j a_k a_l.\n\\end{equation}\nThe coefficients $h_{ij}$ and $h_{ijkl}$ are one- and two-electron overlap integrals, which can be precomputed classically and input to the quantum simulation as parameters~\\cite{Alan,James, Mcweeny}. \n\nAs an application of the techniques presented in this paper (Section~\\ref{molham}), we treat molecular hydrogen in a minimal basis. Thus, we construct two spatial molecular orbitals by taking linear combinations of the localized atomic spatial wavefunctions: $\\psi_g = \\psi_{H1} + \\psi_{H2}$ and $\\psi_u = \\psi_{H1} - \\psi_{H2}$. Here the subscripts {\\it g} and {\\it u} stand for the German words {\\it gerade} and {\\it ungerade} --- even and odd. In general one must take a Slater determinant to determine the correctly anti-symmetrized wavefunctions of the fermionic system, but in this case we can guess them by inspection. The form of the spatial wavefunctions is determined by the choice of basis set. STO-3G is a commonly used Gaussian basis set --- for further details see \\cite{Mcweeny,Szabo}.\n\nMolecular spin orbitals are formed by taking the product of these two molecular spatial orbitals with one of two orthogonal spin functions, $\\ket{\\alpha}$ and $\\ket{\\beta}$. Thus, the four molecular spin orbitals in our model of the hydrogen molecule (which correspond to the operators $a_j^{(\\dag)}$) are:\n\\begin{equation}\n\\ket{\\chi_0} = \\ket{\\psi_g} \\ket{\\alpha}, \\qquad \\ket{\\chi_1} = \\ket{\\psi_g} \\ket{\\beta}, \\qquad \\ket{\\chi_2} = \\ket{\\psi_u} \\ket{\\alpha}, \\qquad \\ket{\\chi_3} = \\ket{\\psi_u} \\ket{\\beta}.\n\\end{equation}\nIn the next section we will review the occupation number basis and the Jordan-Wigner transformation, which together have been established as\na standard method for mapping fermionic systems to quantum computers~\\cite{Alan,Somma,James,Lanyon}.\n\n\\subsection{The Jordan-Wigner transformation}\nThe form of electronic occupation number basis vectors suggests the following identification between \nelectronic basis states on the left and states of our quantum computer~\\cite{Zanardi}:\n\\begin{equation} \\ket{f_{n-1}\\ \\ldots\\ f_1\\ f_{0}} \\rightarrow\n\\ket{q_{n-1}} \\cdots \\otimes \\ket{q_1}\\otimes \\ket{q_0}, \\qquad\\ f_j = q_j \\in \\{0,1\\}.\n\\end{equation}\nThat is, we let the state of each qubit $\\ket{q_j}$ store $f_j$, the occupation number of orbital $j$. We refer to this method of encoding fermionic states as the occupation number basis for qubits. The next step is to map fermionic creation and annihilation operators onto operators on qubits.\n\nWe can form one-qubit creation and annihilation operators, $\\hat{Q}^{+}$ and $\\hat{Q}^{-}$, that act on qubits of our quantum computer as follows:\n\\begin{eqnarray}\n& \\hat{Q}^{+}\\ket{0}=\\ket{1}, \\qquad \\hat{Q}^{+}\\ket{1}=0, \\qquad \\hat{Q}^{-} \\ket{1}=\\ket{0}, \\qquad \\hat{Q}^{-} \\ket{0}=0.\n\\end{eqnarray}\nWe could proceed by following the standard recipe for turning $p$-qubit quantum gates into operators acting on an $n$-qubit \nquantum computer ($n \\geq p$) by taking the tensor product of the gates acting on the target qubits with the identity\nacting on the other ($n-p$) qubits. However, it is easy to show that the qubit creation and annihilation operators formed in this way\ndo not obey the fermionic anti-commutation relations.\n\nExpressing the qubit creation and annihilation operators in terms of Pauli matrices suggests a way forward:\n\\begin{equation}\n\\hat{Q}^{+} = \\ket{1}\\bra{0} = \\frac{1}{2}(\\sigma^x - i\\sigma^y), \\qquad\n\\hat{Q}^{-} = \\ket{0}\\bra{1} = \\frac{1}{2}(\\sigma^x + i\\sigma^y).\n\\end{equation}\nThe mutual anti-commutation of the three Pauli matrices allows us to recognize that $\\hat{Q}^{\\pm}$ anti-commutes with $\\sigma^z$. Thus if we represent the action of $a^\\dag_j$ or $a_j$ by acting with $\\hat{Q}^{\\pm}_{j}$ and with $\\sigma^z$ on all qubits with index less than $j$, our qubit operators will obey the fermionic anti-commutation relations. Put differently, the states of our quantum computer will acquire the same phases under the action of our qubit operator as do the electronic basis states under the action of the corresponding creation or annihilation operator. The effect of the string of $\\sigma^z$ gates is to introduce the required phase change of $-1$ if the parity of the set of qubits with index less than $j$ is 1 (odd), and to do nothing if the parity is 0 (even), where the parity of a set of qubits is just the sum ($\\bmod ~2$) of the numbers that represent the states they are in.\n\nWe can then completely represent the fermionic creation and annihilation operators in terms of basic qubit gates as follows:\n\\begin{equation}\n a_j^\\dagger\\equiv{\\mathbf{1}}^{\\otimes n-j-1}\\otimes \\hat{Q}^{+} \\otimes [{\\sigma^z}^{\\otimes j}], \\qquad a_j\\equiv{\\mathbf{1}}^{\\otimes n-j-1}\\otimes \\hat{Q}^{-} \\otimes [{\\sigma^{z}}^{\\otimes j}].\n\\end{equation}\nA more compact notation, of which we will make extensive use throughout this paper, is:\n\\begin{eqnarray}\n&a_j^\\dagger \\equiv \\hat{Q}^{+}_j \\otimes Z^{\\rightarrow}_{j-1} = \\frac{1}{2} (X_j \\otimes Z^{\\rightarrow}_{j-1} - i Y_j \\otimes Z^{\\rightarrow}_{j-1}); \\\\\n&a_j \\equiv \\hat{Q}^{-}_j \\otimes Z^{\\rightarrow}_{j-1} = \\frac{1}{2} (X_j \\otimes Z^{\\rightarrow}_{j-1} + i Y_j \\otimes Z^{\\rightarrow}_{j-1}),\n\\end{eqnarray}\nwhere:\n\\begin{equation} \nZ^{\\rightarrow}_{i} \\equiv \\sigma^z_i \\otimes \\sigma^z_{i-1} \\otimes \\cdots \\sigma^z_1 \\otimes \\sigma^z_0,\n\\end{equation}\nand where it is assumed that any qubit not explicitly operated on is acted on by the identity. The operator $Z^{\\rightarrow}_i$ is a ``parity operator\" with eigenvalues $\\pm 1$, corresponding to eigenstates for which the subset of bits with index less than or equal to $i$ has even or odd parity, respectively.\n\nThe above correspondence, a mapping of interacting fermions to spins, is the Jordan-Wigner transformation \\cite{Alan,JW,James, Aeppli}. Jordan and Wigner introduced this transformation in 1928 in the context of 1D lattice models, but it has since been applied to quantum simulation of fermions \\cite{Alan,Somma, JW, James}. The problem with this method is that as a consequence of the non-locality of the parity operator $Z^{\\rightarrow}_i$, the number of extra qubit operations required to simulate a single fermionic operator scales as $O(n)$. In the next section we consider two alternatives to the occupation number basis that were suggested by Bravyi and Kitaev \\cite{BK}.\n\n\n\\section{Alternatives to the occupation number basis}\\label{alt}\n\\subsection{The parity basis}\nThe extra qubit operations required to simulate one fermionic operator when using the Jordan-Wigner method result from operating with $\\sigma^z$ on all qubits with index less than $j$. This task could be accomplished by a single application of $\\sigma^z$ if instead of using qubit $j$ to store $f_j$, we used qubit $j$ to store the {\\em parity} of all occupied orbitals up to orbital $j$ \\cite{BK}. That is, we could let qubit $j$ store $p_j=\\sum_{s=0}^{j}f_s$. (Throughout this paper, all sums of binary variables are taken $\\bmod ~2$). We follow~\\cite{BK} and call this encoding of fermionic states in qubit states the {\\em{parity}} basis. \n\nIt is useful to define the transformations between bases we will consider in terms of maps between bit strings. For all the transformations we consider, which involve only sums of bits $\\bmod ~2$, it is possible to represent their action by matrices acting on the vector of bit values corresponding to a given logical basis state. For example, the occupation number basis state $\\ket{f_7 \\ldots f_1f_0}$ is equivalent to the following vector:\n\n\\begin{equation}\n(f_7, \\dots, f_1, f_0)^T \\\\\n\\end{equation}\n\nIn terms of these vectors the map to the parity basis is given by:\n\\begin{equation}\\label{threeone}\np_i = \\sum_j [ \\pi_n]_{ij}\\ f_j,\n\\end{equation}\nwhere $n$ is the number of orbitals. $\\pi_n$ is the $(n\\times n)$ matrix defined below. Note that we index the matrix $\\pi_n$ from the lower right corner, for consistency with our orbital numbering scheme. \n\\begin{equation}\n [\\pi_n]_{ij} = \\left\\{ \n \\begin{array}{l l}\n 1 & \\quad i < j \\\\\n 0 & \\quad i \\geq j\\\\\n \\end{array} \\right. ,\n \\qquad\n {\\rm so~that}\n \\qquad\n \\pi_n = \\left(\n \\begin{array}{cccc}\n 1 & 1 & \\cdots & 1 \\\\\n 0 & 1 & \\cdots & 1 \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & \\cdots & 1\n \\end{array}\n \\right)\n\\end{equation}\nFor example, to change the occupation number basis state $\\ket{1 0 1 0 0 1 1 1}$ into its corresponding parity basis state $\\ket{1 0 0 1 1 1 0 1}$, we act with the matrix $\\pi_8$ on the appropriate bit string:\n\\begin{equation}\n\\begin{bordermatrix}{~ & f_7 & f_6 & f_5 & f_4 & f_3 & f_2 & f_1 & f_0 \\cr\n p_7 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\cr\n p_6 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\cr\n p_5 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\cr\n p_4 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\cr\n p_3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\cr\n p_2 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\cr\n p_1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\cr\n p_0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\cr}\n \\end{bordermatrix}\n\\left(\\begin{matrix}\n {1 \\cr\n 0 \\cr\n 1 \\cr\n 0 \\cr\n 0 \\cr\n 1 \\cr\n 1 \\cr\n 1\\cr}\n \\end{matrix}\n \\right)\n =\n \\begin{pmatrix}\n { 1 \\cr\n 0 \\cr\n 0 \\cr\n 1 \\cr\n 1 \\cr\n 1 \\cr\n 0 \\cr\n 1\\cr}\n \\end{pmatrix}\n\\end{equation}\nWith this understanding of the parity basis transformation, we can now derive the transformation that maps fermionic operators into operators in the parity basis. Since the parity of the set of orbitals with index less than $j$ is what determines whether the action of $a^{(\\dag)}_j$ introduces a phase of $-1$, operating with $\\sigma^z$ on qubit $(j-1)$ alone will introduce the necessary phase to the corresponding qubit state in the parity basis.\n\nHowever, unlike the Jordan-Wigner transformation, we cannot represent the creation or annihilation of a particle in orbital $j$ by simply operating\nwith $\\hat{Q}^{\\pm}$ on qubit $j$, because in the parity basis qubit $j$ does not store the occupation of orbital $j$, but the parity of all\norbitals with index less than or equal to $j$. Thus whether we need to act with $\\hat{Q}^{+}$ or $\\hat{Q}^{-}$ on qubit $j$\ndepends on qubit $(j-1)$. If qubit $(j-1)$ is in the state $\\ket{0}$, then qubit $j$ will accurately reflect the\noccupation of orbital $j$, and simulating $a^\\dag_j$ will require acting on qubit $j$ with $\\hat{Q}^{+}$, as before. But if qubit $(j-1)$ is in the state $\\ket{1}$,\nthen qubit $j$ will have inverted parity compared to the occupation of orbital $j$, and we will instead need to act with $\\hat{Q}^{-}$ on qubit $j$ to simulate $a^\\dag_j$ (and {\\em vice versa} for the annihilation operator).\n\nThe operator equivalent to $\\hat{Q}^{\\pm}$ in the parity basis is therefore a two-qubit operator acting on qubits $j$ and $j-1$:\n\\begin{equation}\n\\hat{\\mathcal{P}}^{\\pm}_j \\equiv \\hat{Q}^{\\pm}_j \\otimes \\ket{0}\\bra{0}_{j-1} - \\hat{Q}^{\\mp}_j \\otimes \\ket{1}\\bra{1}_{j-1} = \\frac{1}{2}(X_j \\otimes Z_{j-1} \\mp i Y_j).\n\\end{equation}\nAdditionally, creating or annihilating a particle in orbital $j$ changes the parity data that must be stored by\nall qubits with index greater than $j$. Thus we must update the cumulative sums $p_k$ for $k > j$ by applying\n$\\sigma^x$ to all qubits $\\ket{p_k}$, $k > j$ \\cite{BK}. The representations of the creation and annihilation operators in the parity basis are then:\n\n\\begin{eqnarray}\n&a_j^\\dagger \\equiv X^{\\leftarrow}_{j+1} \\otimes \\hat{\\mathcal{P}}^{+}_j = \\frac{1}{2} (X^{\\leftarrow}_{j+1} \\otimes X_j \\otimes Z_{j-1} - i X^{\\leftarrow}_{j+1} \\otimes Y_j); \\\\\n&a_j \\equiv X^{\\leftarrow}_{j+1} \\otimes \\hat{\\mathcal{P}}^{-}_j = \\frac{1}{2} (X^{\\leftarrow}_{j+1} \\otimes X_j \\otimes Z_{j-1} + i X^{\\leftarrow}_{j+1} \\otimes Y_j),\n\\end{eqnarray}\nwhere:\n\\begin{equation}\nX^{\\leftarrow}_{i} \\equiv \\sigma^x_{n-1} \\otimes \\sigma^x_{n-2} \\otimes \\cdots \\sigma^x_{i+1} \\otimes \\sigma^x_{i}.\n\\end{equation}\nThis is the equivalent of the Jordan-Wigner transformation for the parity basis. The operator $X^{\\leftarrow}_{i}$ is the ``update operator\", which updates all qubits that store a partial sum including orbital $(i-1)$ when the occupation number of that orbital changes. It is straightforward to verify that these mappings satisfy the fermionic anti-commutation relations. But to simulate fermionic operators in the parity basis, we have traded the trailing string of $\\sigma^z$ gates required by the\nJordan-Wigner transformation for a leading string of $\\sigma^x$ gates whose length also scales as $O(n)$, and we have\nnot improved on the efficiency of the Jordan-Wigner simulation procedure. In the next section, we explore a third possibility.\n\n\\subsection{The Bravyi-Kitaev basis}\nTwo kinds of information are required to simulate fermionic operators with qubits: the occupation of the target orbital, and the parity of the set of orbitals with index less than the target orbital. The previous two approaches are dual in the way that they store this information. With the occupation number basis and its associated Jordan-Wigner transformation, the occupation information is stored locally but the parity information is non-local, whereas in the parity basis method and its corresponding operator transformation, the parity information is stored locally but the occupation information is non-local. \n\nThe Bravyi-Kitaev basis is a middle ground.\nThat is, it balances the locality of occupation and parity information for improved simulation efficiency. The general form of such a scheme\nmust be to use qubits $\\ket{b_j}$ to store $partial$ sums $\\sum_{s=k}^{l}f_s$ of occupation numbers according to some\nalgorithm. For ease of explanation, in the exposition that follows, when we write that a qubit ``stores a set of orbitals\", what is meant is that the qubit\nstores the parity of the set of occupation numbers corresponding to that set of orbitals.\n\nBravyi and Kitaev's encoding has an elegant binary grouping structure \\cite{BK}. In this scheme, qubits store the parity of a set of $2^x$ orbitals, where $x \\geq 0$. A qubit of index $j$ always stores orbital $j$. For even values of $j$, this is the only orbital that it stores, but for odd values of $j$, it also stores a certain set of adjacent orbitals with index less than $j$. Just as with the parity basis transformation, this encoding can be symbolized in a matrix $\\beta_n$ that acts on bit string vectors corresponding to occupation number basis vectors of length $n$ to transform them to the corresponding Bravyi-Kitaev-encoded bit strings (again, all additions done ${\\bmod \\ 2}$). In terms of these vectors, the map from the occupation number basis to the Bravyi-Kitaev basis is:\n\\begin{equation}\nb_i = \\sum_j [ \\beta_n]_{ij}\\ f_j,\n\\end{equation}\nwhere the matrix $\\beta_n$ is given in Figure~\\ref{figtwo} below.\n\n\\begin{figure}[!ht]\n \\centerline{\\includegraphics[scale = .5]{figure2a.png}}\n \\centerline{\\includegraphics[scale = .5]{figure2b.png}}\n \\caption[Change of basis matrix (from occupation to Bravyi-Kitaev)]{\\label{figtwo}The matrix $\\beta_n$ that transforms occupation number basis vectors of length $n$ into the Bravyi-Kitaev basis. $\\beta_1$ is a $(1 \\times 1)$ matrix with a single entry of 1. Subsequent iterations of the matrix that act on occupation number basis vectors\nof length $2^x$ are constructed by taking $\\mathbf{1} \\otimes \\beta_{2^{x-1}}$ and then filling in the top row of the first quadrant of this matrix with 1's.\n$\\beta_n$ for $2^x < n < 2^{x+1}$ is just the $(n \\times n)$ segment of $\\beta_{2^{x+1}}$ that includes $b_0$ through\n$b_{n-1}$. The recursion pattern for the inverse transformation matrix is also shown. An entry of 1 in row $b_i$, column $f_j$ means that $b_i$ is a partial sum including $f_j$.}\n\n\\end{figure}\n\nFor example, to change the occupation number basis state $\\ket{1 0 1 0 0 1 1 1}$ into its corresponding Bravyi-Kitaev basis state $\\ket{1 0 1 0 1 1 0 1}$, we act with the matrix $\\beta_8$ on the appropriate bit string vector:\n\\begin{equation}\n\\begin{bordermatrix}{~ & f_7 & f_6 & f_5 & f_4 & f_3 & f_2 & f_1 & f_0 \\cr\n p_7 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\cr\n p_6 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\cr\n p_5 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\cr\n p_4 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\cr\n p_3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\cr\n p_2 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\cr\n p_1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\cr\n p_0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\cr}\n \\end{bordermatrix}\n\\left(\\begin{matrix}\n {1 \\cr\n 0 \\cr\n 1 \\cr\n 0 \\cr\n 0 \\cr\n 1 \\cr\n 1 \\cr\n 1\\cr}\n \\end{matrix}\n \\right)\n =\n \\left(\\begin{matrix}\n { 1 \\cr\n 0 \\cr\n 1 \\cr\n 0 \\cr\n 1 \\cr\n 1 \\cr\n 0 \\cr\n 1\\cr}\n \\end{matrix}\n \\right)\n\\end{equation}\n\n\nThis encoding strikes a balance between the occupation number basis and the parity basis\nmethods. The parity of occupied orbitals up to orbital $j$ is no longer stored in a single qubit, but the Bravyi-Kitaev\nencoding stores the parity of orbitals with index less than $j$ in a few partial sums whose number scales as $O(\\log j) \\leq O(\\log n)$ \\cite{BK}.\nLikewise, we no longer need to update {\\em all} the qubits with index\ngreater than $j$, but only those that store partial sums which include occupation number $j$. Each\noccupation number enters an additional partial sum only if the number of single particle states $n$ is doubled, and so\nthe overall cost of simulating a single fermionic operator with qubits scales as $O(\\log n)$ \\cite{BK}.\n\nGiven this encoding, we need to determine --- for an arbitrary index $j$ --- which qubits in the Bravyi-Kitaev basis store the parity of all orbitals with index less than $j$, which qubits store a partial sum including orbital $j$, and which qubits determine whether qubit $j$ has the same parity or inverted parity with respect to orbital $j$. These sets of indices will allow us to explicitly construct the fermionic creation and annihilation operators in the Bravyi-Kitaev basis. In the next section, we define these sets of qubit indices.\n\n\n\\section{Sets of qubits relevant to the Bravyi-Kitaev basis}\\label{sets}\nIn this section we define the sets of qubits that are involved in the Bravyi-Kitaev transformation. These are the {\\em parity set} (the qubits in the Bravyi-Kitaev basis that store the parity of all orbitals with index less than $j$), the {\\em update set} (the qubits that store a partial sum including orbital $j$), and the {\\em flip set} (the qubits that determine whether qubit $j$ has the same parity as orbital $j$).\n\n\\subsection{The parity set}\nFor an arbitrary index $j$, we would like to know which set of qubits in the Bravyi-Kitaev basis tells us whether or not the state of the quantum computer needs to acquire a phase change of $-1$ under the action of a creation or annihilation operator acting on orbital $j$. The parity of this set of qubits has the same parity as the set of orbitals with index less than $j$, and so we will call this set of qubit indices the ``parity set\" of index $j$, or $P(j)$. To determine the elements of $P(j)$, we consider the transformation from the Bravyi-Kitaev basis to the parity basis. From equation~(\\ref{threeone}) we know that $p_i = \\sum_j [ \\pi_n]_{ij}\\ f_j$. Given the inverse transformation matrix $\\beta_n^{-1}$, it is also true that:\n\\begin{equation}\nf_j = \\sum_k [ \\beta_n^{-1}]_{jk}\\ b_k,\n\\end{equation}\n\\noindent and hence:\n\\begin{eqnarray}\np_i &= \\sum_j [ \\pi_n]_{ij}\\ (\\sum_k [ \\beta_n^{-1}]_{jk}\\ b_k)\\\\\n &= \\sum_k [\\pi_n \\beta_n^{-1}]_{ik}\\ b_k \n\\end{eqnarray}\n\nThe matrix $\\pi_n \\beta_n^{-1}$ is the transformation matrix from the Bravyi-Kitaev basis to the parity basis. Therefore, the nonzero entries to the right of the main diagonal in row $i$ of the matrix $\\pi_n \\beta_n^{-1}$ give the indices of qubits in the Bravyi-Kitaev basis that can be used to compute the cumulative parity of orbitals with index less than $i$. An entry of 1 in row $i$, column $j$ of $\\pi_n \\beta_n^{-1}$ (where $j < i$, i.e. to the right of the main diagonal by our numbering) indicates that $j \\in P(i)$:\n\\begin{equation}\n\\pi_8 \\beta_8^{-1} =\n\\bordermatrix{~ & _7 & _6 & _5 & _4 & _3 & _2 & _1 & _0 \\cr\n _7 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\cr\n _6 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\cr\n _5 & 0 & 0 & 1& 1 & 1 & 0 & 0 & 0 \\cr\n _4 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\cr\n _3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\cr\n _2 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\cr\n _1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\cr\n _0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\cr}\n \\qquad\n {\\rm which~implies:}\n \\qquad\n \\left\\{ \n \\begin{array}{l}\n P(7) = \\{6,5,3\\}\\\\\n P(6) = \\{5,3\\}\\\\\n P(5) = \\{4,3\\}\\\\\n P(4) = \\{3\\}\\\\\n P(3) = \\{2,1\\}\\\\\n P(2) = \\{1\\}\\\\\n P(1) = \\{0\\}\\\\\n P(0) = \\emptyset \\\\\n \\end{array} \\right.\n\\end{equation}\n\n\\subsection{The update set}\n\nFor arbitrary $j$, we define the set of qubits (other than qubit $j$) that must be updated when the occupation of orbital $j$ changes. We call this set the ``update set\" of index $j$, or $U(j)$. This is the set of qubits in the Bravyi-Kitaev basis that store a partial sum including orbital $j$. Any Bravyi-Kitaev qubit that stores a partial sum that includes occupation number $j$ is in $U(j)$. Since even indexed qubits store only the occupation of the corresponding orbital, update sets contain only odd indices. It is straightforward to determine the elements of $U(j)$ from the transformation matrix $\\beta_n$ that maps bit strings in the occupation number basis to the Bravyi-Kitaev basis. The columns of this transformation matrix show which qubits in the Bravyi-Kitaev basis store a particular orbital, and so the nonzero entries in column $j$ above the main diagonal determine the qubits other than qubit $j$ that must be updated when the occupancy of orbital $j$ changes. These are the elements of the update set.\n\n\\begin{equation}\n\\beta_8 =\n\\bordermatrix{~ & f_7 & f_6 & f_5 & f_4 & f_3 & f_2 & f_1 & f_0 \\cr\n b_7 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\cr\n b_6 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\cr\n b_5 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\cr\n b_4 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\cr\n b_3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\cr\n b_2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\cr\n b_1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\cr\n b_0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\cr}\n \\qquad\n {\\rm which~implies:}\n \\qquad\n \\left\\{ \n \\begin{array}{l}\n U(7) = \\emptyset\\\\\n U(6) = \\{7\\}\\\\\n U(5) = \\{7\\}\\\\\n U(4) = \\{5,7\\}\\\\\n U(3) = \\{7\\}\\\\\n U(2) = \\{3,7\\}\\\\\n U(1) = \\{3,7\\}\\\\\n U(0) = \\{1,3,7\\} \\\\\n \\end{array} \\right.\n\\end{equation}\n\nIt should be clear that update sets depend on the size of the basis used. For example, if 16 basis functions were used instead of the 8 used in the example above, all the update sets other than $U(7)$ would also include index 15.\n\n\\subsection{The flip set}\nFor arbitrary $j$, we need to know what set of Bravyi-Kitaev qubits determines whether qubit $j$ has the same parity or inverted parity with respect to orbital $j$.\nWe will call this set of Bravyi-Kitaev qubits the ``flip set\" of $j$, or $F(j)$, because this set is responsible for whether $b_j$ has flipped parity with respect to $f_j$. This is the set that stores the parity of occupation numbers other than $f_j$ in the sum $b_j$. Since even-indexed qubits store only the orbital with the same index, the flip set of even indices is always the empty set. One can determine the elements of $F(j)$ by looking at the inverse transformation matrix $\\beta_n^{-1}$ that maps bit strings in the Bravyi-Kitaev basis to the occupation number basis. The columns with nonzero entries to the right of the main diagonal in row $i$ of this inverse transformation matrix give the indices of the Bravyi-Kitaev qubits that together store the same set of orbitals as is stored by $\\ket{b_i}$. These are the elements of the flip set.\n\n\\begin{equation}\n\\beta_8^{-1} =\n\\bordermatrix{~ & b_7 & b_6 & b_5 & b_4 & b_3 & b_2 & b_1 & b_0 \\cr\n f_7 & 1 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\cr\n f_6 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\cr\n f_5 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\cr\n f_4 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\cr\n f_3 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\cr\n f_2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\cr\n f_1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\cr\n f_0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\cr}\n \\qquad\n {\\rm which~implies:}\n \\qquad\n \\left\\{ \n \\begin{array}{l}\n F(7) = \\{6,5,3\\} \\\\\n F(6) = \\emptyset \\\\\n F(5) = \\{4\\}\\\\\n F(4) = \\emptyset \\\\\n F(3) = \\{2,1\\}\\\\\n F(2) = \\emptyset \\\\\n F(1) = \\{0\\} \\\\\n F(0) = \\emptyset \\\\\n \\end{array} \\right.\n\\end{equation}\nWith these sets defined, we can derive the mapping from fermionic operators to qubit operators that is the equivalent of the Jordan-Wigner transformation in the Bravyi-Kitaev basis.\n\n\n\\section{The Bravyi-Kitaev transformation}\\label{bravkit}\nIn this section we will give an explicit prescription, in terms of Pauli matrices, for representing the creation and\nannihilation operators that act on the Bravyi-Kitaev basis states. Operating in this basis requires that we find the analogues to the qubit creation and annihilation operators \n($\\hat{Q}^{\\pm}$ in the occupation number basis, $\\hat{\\mathcal{P}}^{\\pm}$ in the parity basis) as well as the\nparity operator, $Z^{\\rightarrow}_{i}$, and the update operator, $X^{\\leftarrow}_{i}$, in the Bravyi-Kitaev basis.\nWe will first define some notation.\n\nFor our purposes it is the parity of subsets of orbitals or qubits that matters, not the individual\noccupation numbers or states of the qubits in the set. Thus, it is useful to define operators that project onto the subspace of the Hilbert space of the entire computer for which the subset of qubits with indices in $S$ has the parity selected for by the operator (even for ${\\hat{E}}_S$, odd for ${\\hat{O}}_S$). We can express these operators in terms of Pauli matrices as follows:\n\\begin{equation}\\label{eq:5.2}\n\\hat{E}_S = \\frac{1}{2}(\\mathbf{1} + Z_S), \\qquad {\\hat{O}}_S = \\frac{1}{2}(\\mathbf{1} - Z_S),\n\\end{equation}\nwhere $Z_S$ is shorthand for the $\\sigma^z$ gate applied to all qubits in $S$. With this notation established, we will\nnext write equations for the qubit operators in the Bravyi-Kitaev basis that represent creation and annihilation operators acting on orbital $j$. To begin we will consider the case for which $j$ is even, because this will allow us to build intuition for the more difficult case for which $j$ is odd.\n\n\\subsection{Representing $a_j^{(\\dag)}$ in the Bravyi-Kitaev basis for $j$ even}\n\nIn the case that $j$ is even, we should act with $\\hat{Q}^\\pm$ on qubit $j$, just as for the Jordan-Wigner transformation, because the Bravyi-Kitaev encoding stores orbitals with $j = 0 \\pmod 2$ in the qubit with the same index. There are then two additional tasks that dictate how to represent the fermionic operators in the Bravyi-Kitaev basis: determining the parity of occupied orbitals with index less than $j$, and updating qubits with index greater than $j$ that store a partial sum that includes occupation number $j$.\n\nThe parity of the set of qubits in $P(j)$ is equal to that of the set of orbitals with index less than $j$. By analogy with the Jordan-Wigner transformation, we act with $\\sigma^z$ on all qubits with indices in $P(j)$, that is, we apply the operator $Z_{P(j)}$. The number of qubits in $P(j)$ scales as $O(\\log j) \\leq O(\\log n)$ \\cite{BK}. \n\nSecondly, by analogy with the parity basis method, we also act with $\\sigma^x$ on all qubits in the appropriate $U(j)$; that is, we apply the operator $X_{U(j)}$. This has the effect of updating all the qubits that store a set of orbitals including orbital $j$. The size of $U(j)$ also scales like $O(\\log n)$~\\cite{BK}. To summarize: to represent $a^\\dag_j$ or $a_j$ in the Bravyi-Kitaev basis, for $j$ even, we act with $\\sigma^z$ on all qubits in $P(j)$, $\\hat{Q}^{\\pm}$ on qubit $j$, and with $\\sigma^x$ on all qubits in $U(j)$:\n\\begin{eqnarray}\n&a^\\dag_j \\equiv X_{U(j)} \\otimes \\hat{Q}^{+}_j \\otimes Z_{P(j)} = \\frac{1}{2}(X_{U(j)} \\otimes X_j \\otimes Z_{P(j)} - i X_{U(j)} \\otimes Y_j \\otimes Z_{P(j)}); \\\\\n&a_j \\equiv X_{U(j)} \\otimes \\hat{Q}^{-}_j \\otimes Z_{P(j)} = \\frac{1}{2}(X_{U(j)} \\otimes X_j \\otimes Z_{P(j)} + i X_{U(j)} \\otimes Y_j \\otimes Z_{P(j)}).\n\\end{eqnarray}\nIn the next section, we will consider the case for which $j$ is odd.\n\n\\subsection{Representing $a_j^{(\\dag)}$ in the Bravyi-Kitaev basis for $j$ odd}\n\nTo represent the creation or annihilation of a particle in orbital $j$ in the Bravyi-Kitaev basis, for $j$ even, we could simply act with\n$\\hat{Q}^{\\pm}$ on qubit $j$ because that qubit stores only the occupation of orbital $j$. For $j$ odd, qubit $j$ stores a partial sum of\noccupation numbers of orbitals including, but not limited to, orbital $j$. Thus, in this case the state of Bravyi-Kitaev qubit $j$ is either equal to the occupation of orbital $j$ (if the parity of the other orbitals that it stores is even), or opposite to that of orbital $j$ (if the parity of the other orbitals that it stores is 1). Thus, whether representing the creation or annihilation of a particle in orbital\n$j$ requires that we act with\n$\\hat{Q}^+$ or $\\hat{Q}^-$ on qubit $j$ in the Bravyi-Kitaev basis depends on the parity of all occupation numbers other than $f_j$ that are\nincluded in the partial sum $b_j$ --- i.e. the parity of the flip set of index $j$. If the parity of the set of qubits with indices in $F(j)$ is even, then the creation or annihilation of a particle in orbital $j$ requires acting with $\\hat{Q}^+$\nor $\\hat{Q}^-$, respectively, as usual. But if the parity of this set of qubits is odd, then the creation of a particle requires acting with $\\hat{Q}^-$ and the annihilation of a particle requires acting with $\\hat{Q}^+$. The Bravyi-Kitaev analogues to the qubit creation and annihilation operators are therefore:\n\n\\begin{equation}\\label{eq:5.4}\n\\hat{\\Pi}^{\\pm}_j \\equiv \\hat{Q}^{\\pm}_j \\otimes \\hat{E}_{F(j)} - \\hat{Q}^{\\mp}_j \\otimes \\hat{O}_{F(j)} = \\frac{1}{2}(X_j \\otimes Z_{F(j)} \\mp i Y_j).\n\\end{equation}\n\\noindent The updating procedure in this case in which $j$ is odd works in exactly the same way as it does in the case that $j$ is even. In applying the parity operator, however, we need only consider the qubits that are in $P(j)$ but not in $F(j)$, because the relative sign in the $\\hat{\\Pi}^{\\pm}_j$ operator implicitly calculates the parity of the subset of the parity set that is also in the flip set of index $j$. It is convenient to therefore introduce the new ``remainder set\":\n\\begin{equation}\nR(j) \\equiv P(j) \\setminus F(j).\n\\end{equation}\nThus, the fermionic creation and annihilation operators acting on orbital $j$ for $j$ odd are represented in the Bravyi-Kitaev basis as follows:\n\\begin{eqnarray}\n&a^\\dag_j \\equiv X_{U(j)} \\otimes \\hat{\\Pi}^{+}_j \\otimes Z_{R(j)} = \\frac{1}{2}(X_{U(j)} \\otimes X_j \\otimes Z_{P(j)} - i X_{U(j)} \\otimes Y_j \\otimes Z_{R(j)}); \\\\\n&a_j \\equiv X_{U(j)} \\otimes \\hat{\\Pi}^{-}_j \\otimes Z_{R(j)} = \\frac{1}{2}(X_{U(j)} \\otimes X_j \\otimes Z_{P(j)} + i X_{U(j)} \\otimes Y_j \\otimes Z_{R(j)}).\n\\end{eqnarray}\n\n\n\\noindent It is evident by inspection that the only difference in the algebraic form of the operators between the even- and odd-indexed cases is that the second term involves $Z_{P(j)}$ for the even case, but $Z_{R(j)}$ for the odd case. Therefore we define:\n\n\\begin{equation}\n \\rho(j) \\equiv \\left\\{ \n \\begin{array}{l l}\n P(j) & \\quad {\\rm if~}j{\\rm~is~even;}\\\\\n R(j) & \\quad {\\rm if~}j{\\rm~is~odd.}\\\\\n \\end{array} \\right.\n\\end{equation}\n\n\\noindent Now the fermionic creation and annihilation operators acting on arbitrary $j$ are represented in the Bravyi-Kitaev basis as:\n\n\\begin{eqnarray}\n&a^\\dag_j \\equiv X_{U(j)} \\otimes \\hat{\\Pi}^{+}_j \\otimes Z_{R(j)} = \\frac{1}{2}(X_{U(j)} \\otimes X_j \\otimes Z_{P(j)} - i X_{U(j)} \\otimes Y_j \\otimes Z_{\\rho(j)}); \\\\\n&a_j \\equiv X_{U(j)} \\otimes \\hat{\\Pi}^{-}_j \\otimes Z_{R(j)} = \\frac{1}{2}(X_{U(j)} \\otimes X_j \\otimes Z_{P(j)} + i X_{U(j)} \\otimes Y_j \\otimes Z_{\\rho(j)}).\n\\end{eqnarray}\n\n\\noindent These are useful basic results, but the operators that appear in the molecular electronic Hamiltonian are actually products of these creation and annihilation operators. In the next section, we derive general expressions for products of these second-quantized operators.\n\n\n\\section{Pauli representations of second-quantized operators in the Bravyi-Kitaev basis}\\label{pauli}\n\nIn this Section we derive simplified algebraic expressions for classes of Hermitian second-quantized fermionic operators in the Bravyi-Kitaev basis. The five relevant classes of operators are summarized in Table~\\ref{tab1}. We will give complete compact algebraic expressions for only the number operators and the Coulomb and exchange operators. It is not possible to give the algebraic form for the remaining three classes of operators without considering an impractical number of sub-cases, so we opt to give general expressions for products of the form $a_i^\\dag a_j$, and show how to use these results to generate algebraic expressions for the remaining classes of operators.\n\n\n\\begin{table}[!h]\n\\begin{center}\n\\begin{tabular}{ |c | c| }\n \\hline\n {\\bf Operator} & {\\bf Second quantized form} \\\\ \n\n \\hline\nNumber operator & $h_{ii}\\ a^\\dag_i a_i$ \\\\\n\\hline\nCoulomb\/exchange operators & $h_{ijji} \\ a^\\dag_i a^\\dag_j a_j a_i$ \\\\\n\\hline\nExcitation operator & $h_{ij} \\ (a^\\dag_i a_j + a^\\dag_j a_i)$ \\\\ \n\\hline\nNumber-excitation operator & $h_{ijjk} \\ (a^\\dag_i a^\\dag_j a_j a_k + a^\\dag_k a^\\dag_j a_j a_i)$ \\\\ \n\\hline\nDouble excitation operator & $h_{ijkl} \\ (a^\\dag_i a^\\dag_j a_k a_l + a^\\dag_l a^\\dag_k a_j a_i)$ \\\\\n\\hline\n\\end{tabular}\n\\caption{The five classes of Hermitian second quantized operators that appear in electronic Hamiltonians. In general the overlap integrals $h_{ij}$ and $h_{ijkl}$ may be complex.\\label{tab1}}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Number operators: $h_{ii}\\ a^\\dag_i a_i$ }\\label{sixone}\nThe number operators are of the form $h_{ii}\\ a^\\dag_i a_i$ and have eigenvalues corresponding to the occupation number of orbital $i$. We would like to find a simplified expression for this class of operators in the Bravyi-Kitaev basis. \n\nGiven the results of Section~\\ref{bravkit}, we can write the following:\n\\begin{eqnarray}\na^\\dag_i a_i = \\ &\\frac{1}{2}(X_{U(i)} \\otimes X_i \\otimes Z_{P(i)} - i X_{U(i)} \\otimes Y_i \\otimes Z_{\\rho(i)}) \\\\\n\\times &\\frac{1}{2}(X_{U(i)} \\otimes X_i \\otimes Z_{P(i)} + i X_{U(i)} \\otimes Y_i \\otimes Z_{\\rho(i)}). \\nonumber\n\\end{eqnarray}\nGiven that $\\sigma^x \\sigma^x = \\sigma^y \\sigma^y = \\sigma^z \\sigma^z = \\mathbf{1}$, it follows that $(X_S)^2 = (Y_S)^2 = (Z_S)^2 = \\mathbf{1}$. We are left with:\n\\begin{eqnarray}\na^\\dag_i a_i &= \\frac{1}{4}[\\mathbf{1} + i (X_i Y_i) \\otimes Z_{P(i)\\setminus \\rho(i)} - i (Y_i X_i) \\otimes Z_{P(i)\\setminus \\rho(i)} + \\mathbf{1}] \\\\\n &= \\frac{1}{2}(\\mathbf{1} - Z_i \\otimes Z_{P(i)\\setminus \\rho(i)}).\n\\end{eqnarray}\nNow, when $i$ is even, $\\rho(i) = P(i)$, and so $P(i) \\setminus \\rho(i) = \\emptyset$. When $i$ is odd, $\\rho(i) = R(i)$, and so $P(i) \\setminus \\rho(i) = F(i)$. Conveniently, $F(i) = \\emptyset$ for $i$ even, so if we define the following:\n\\begin{equation}\n\\underline{F(i)} \\equiv F(i) \\cup \\{i\\},\n\\end{equation}\nthen we can represent the number operators for arbitrary $i$ (even or odd) as follows:\n\\begin{equation}\na^\\dag_i a_i = \\frac{1}{2}(\\mathbf{1} - Z_{\\underline{F(i)}}).\n\\end{equation}\nIn the next section we consider the Coulomb and exchange operators.\n\n\\subsection{Coulomb and exchange operators: $h_{ijji}\\ a^\\dag_i a_j^\\dag a_j a_i$}\n\nThe Coulomb operators are of the form $a^\\dag_i a_j^\\dag a_j a_i$, while the exchange operators are of the form $a^\\dag_i a_j^\\dag a_i a_j = - a^\\dag_i a_j^\\dag a_j a_i$. Since these two kinds of operators can be grouped together algebraically, we consider them as one case. The fermionic anti commutation relations ensure that $a^\\dag_i a_j^\\dag a_j a_i = - a^\\dag_i a_j^\\dag a_i a_j = (a^\\dag_i a_i)(a_j^\\dag a_j)$. Thus, we can consider the Coulomb and exchange operators as a product of two number operators. With the result from Section~\\ref{sixone}, we can write the following:\n\n\\begin{eqnarray}\n a^\\dag_i a_j^\\dag a_j a_i = \\ &\\frac{1}{2}(\\mathbf{1} - Z_{\\underline{F(i)}}) \\times \\frac{1}{2}(\\mathbf{1} - Z_{\\underline{F(j)}}) \\\\\n = \\ &\\frac{1}{4}(\\mathbf{1} - Z_{\\underline{F(i)}} - Z_{\\underline{F(j)}} + Z_{\\underline{F(i)}} Z_{\\underline{F(j)}}).\n \\end{eqnarray}\nAny overlap between supp($Z_{\\underline{F(i)}}$) and supp($Z_{\\underline{F(j)}}$), where supp($\\hat{O}$) is the support of the operator $\\hat{O}$, i.e. those tensor factors on which it acts nontrivially, will result in the local product $\\sigma^z \\sigma^z = \\mathbf{1}$. Thus, we only actually need to act with $\\sigma^z$ on the union of $\\underline{F(i)}$ and $\\underline{F(j)}$ minus their intersection, i.e. the symmetric difference of these two sets. Thus we define the following notation:\n\\begin{equation}\n\\underline{F_{ij}} \\equiv \\underline{F(i)} \\bigtriangleup \\underline{F(j)} = (\\underline{F(i)} \\cup \\underline{F(j)}) \\setminus (\\underline{F(i)} \\cap \\underline{F(j)}).\n\\end{equation}\nWe can then give the algebraic expression for the Coulomb and exchange operators:\n\\begin{equation}\na^\\dag_i a_j^\\dag a_j a_i = \\frac{1}{4}(\\mathbf{1} - Z_{\\underline{F(i)}} - Z_{\\underline{F(j)}} + Z_{\\underline{F_{ij}}}).\n\\end{equation}\nIn the next section we consider general products of the form $a_i^\\dag a_j$.\n\n\\subsection{Products of the form $a^\\dag_i a_j$}\n\nWe can assume without loss of generality that $i < j$. The algebraic form for products of this kind depends on the parity of the indices. There are four cases and we will work through the first case in detail, and simply present the results for the other cases.\n\nUsing the result of Section~\\ref{bravkit}, we obtain the following when $i$ and $j$ are even:\n\\begin{eqnarray}\\label{sixeight}\na_i^\\dag a_j = \\ &\\frac{1}{2}(X_{U(i)} \\otimes X_i \\otimes Z_{P(i)} - i X_{U(i)} \\otimes Y_i \\otimes Z_{P(i)}) \\\\\n\\times \\ &\\frac{1}{2}(X_{U(j)} \\otimes X_j \\otimes Z_{P(j)} + i X_{U(j)} \\otimes Y_j \\otimes Z_{P(j)}). \\nonumber\n\\end{eqnarray}\n\nFor each of the four terms resulting from multiplying out the operators in equation~(\\ref{sixeight}) above, we must consider what products of local qubit operators can result. There are three potential sources of local qubit operator products: overlap between the update set of qubit $i$ and the update set of qubit $j$, overlap between the update set of qubit $i$ and the parity set of qubit $j$, and overlap between the parity set of qubit $i$ and the parity set of qubit $j$. Any overlap between the update sets of qubits $i$ and $j$ will result in the local product $\\sigma^x \\sigma^x = \\mathbf{1}$; any overlap between the update set of qubit $i$ and parity set of qubit $j$ will result in the local product $\\pm i \n\\sigma^y$; and any overlap in the parity sets of qubits $i$ and $j$ will result in the local product $\\sigma^z \\sigma^z = \\mathbf{1}$. Thus we define the following sets:\n\\begin{equation}\nU_{ij} \\equiv U(i) \\bigtriangleup U(j), \\quad \\quad \\alpha_{ij} \\equiv U(i) \\cap P(j), \\quad \\quad P_{ij}^0 \\equiv P(i) \\bigtriangleup P(j).\n\\end{equation}\nNote that in the case that $i$ and $j$ are even, we do not need to consider the possibility that $j \\in U(i)$ because $U(i)$ contains only odd elements. Similarly, we do not need to consider the possibility that $i \\in P(j)$, because $P(j)$ for $j$ even contains only odd elements.\n\nAs an example, we will show how to use the sets defined above to simplify the term $(X_{U(i)} \\otimes X_i \\otimes Z_{P(i)})(X_{U(j)} \\otimes X_j \\otimes Z_{P(j)})$. For this term, we need only apply $\\sigma^x$ to the set of qubits $U_{ij} \\setminus \\alpha_{ij} \\cup \\{i,j\\}$, $\\sigma^y$ to the qubit with index in $\\alpha_{ij}$ (which set in general has at most 1 element, and in the case that $i$ and $j$ are even always contains 1 element), and $\\sigma^z$ to the qubits in the set $P_{ij}^0 \\setminus \\alpha_{ij}$. Thus, this term simplifies to:\n\\begin{equation}\n(X_{U(i)} \\otimes X_i \\otimes Z_{P(i)})(X_{U(j)} \\otimes X_j \\otimes Z_{P(j)}) = -i \\ X_{U_{ij} \\setminus \\alpha_{ij} \\cup \\{i,j\\}} Y_{\\alpha_{ij}} Z_{P_{ij}^0 \\setminus \\alpha_{ij}}.\n\\end{equation}\nUsing the same reasoning for the other terms, we arrive at the following result:\n \\begin{equation}\na_i^\\dag a_j = \\frac{1}{4} X_{U_{ij} \\setminus \\alpha_{ij}} Y_{\\alpha_{ij}} Z_{P_{ij}^0 \\setminus \\alpha_{ij}} [Y_j X_i - X_j Y_i -i (X_j X_i + Y_j Y_i)].\n\\end{equation}\nThis is our result for the case that $i$ and $j$ are even. The algebraic expressions for the other cases can be derived in the same manner, with the added complication that the expression for the product $a_i^\\dag a_j$ varies, depending on if $i \\in P(j)$ and\/or $j \\in U(i)$. This complication results in a proliferation of sub-cases: two for the case that $i$ is odd and $j$ is even, three for the case that $i$ is even and $j$ is odd, and four for the case that $i$ and $j$ are odd. The only additional sets we need to define are the analogs of $P_{ij}^0$ for when one or both of the indices are odd:\n\\begin{equation}\nP_{ij}^1 \\equiv P(i) \\bigtriangleup R(j), \\quad \\quad P_{ij}^2 \\equiv R(i) \\bigtriangleup P(j), \\quad \\quad P_{ij}^3 \\equiv R(i) \\bigtriangleup R(j).\n\\end{equation}\nThe results for all cases are summarized below in Table~\\ref{tab2}. In the following sub-sections we show how to use the contents of Table~\\ref{tab2} to generate algebraic expressions for the excitation operators, the number-excitation operators, and the double-excitation operators.\n\n\\subsection{Excitation operators: $h_{ij} \\ (a_i^\\dag a_j + a_j^\\dag a_i)$}\nProviding for the possibility that the integral $h_{ij}$ is complex, we can write:\n\\begin{equation}\nh_{ij} \\ (a_i^\\dag a_j + a_j^\\dag a_i) = \\Re\\{h_{ij}\\} (a_i^\\dag a_j + a_j^\\dag a_i) + \\Im\\{h_{ij}\\} (a_i^\\dag a_j - a_j^\\dag a_i).\n\\end{equation}\nApplying this to the case when $i$ and $j$ are even, we find the following:\n\\begin{eqnarray}\nh_{ij} \\ (a_i^\\dag a_j + a_j^\\dag a_i) = \\frac{1}{2} X_{U_{ij} \\setminus \\alpha_{ij}}\\ Y_{\\alpha_{ij}}\\ Z_{P_{ij}^0 \\setminus \\alpha_{ij}} [&\\Re\\{h_{ij}\\}(Y_j X_i - X_j Y_i) \\\\\n + &\\Im\\{h_{ij}\\} (X_j X_i + Y_j Y_i)]. \\nonumber\n\\end{eqnarray}\nSimilar expressions for other cases are easily generated by taking the appropriate form of $a_i^\\dag a_j$ from Table~\\ref{tab2}.\n\n\\subsection{Number-excitation operators: $h_{ijjk} \\ (a^\\dag_i a^\\dag_j a_j a_k + a^\\dag_k a^\\dag_j a_j a_i)$}\n\nDue to the fermionic anti-commutation relations, the following is true:\n\\begin{equation}\na^\\dag_i a^\\dag_j a_j a_k + a^\\dag_k a^\\dag_j a_j a_i = (a_i^\\dag a_k + a_k^\\dag a_i) (a_j^\\dag a_j).\n\\end{equation}\nWe see that this is simply a product of an excitation operator and a number operator. We have previously given algebraic expressions for both of these classes of operators, so it is not difficult to combine them for an expression for the number-excitation operators. Let us consider the example when $i$ and $k$ are even. Then we have the following:\n\\begin{eqnarray}\nh_{ijjk} \\ (a_i^\\dag a_k + a_k^\\dag a_i) a_j^\\dag a_j = &\\frac{1}{2} X_{U_{ik} \\setminus \\alpha_{ik}}\\ Y_{\\alpha_{ik}}\\ Z_{P_{ik}^0 \\setminus \\alpha_{ik}} [\\Re\\{h_{ijjk}\\}(Y_k X_i - X_k Y_i) \\\\\n&+ \\Im\\{h_{ijjk}\\} (X_k X_i + Y_k Y_i)] \\times \\frac{1}{2}(\\mathbf{1} - Z_{\\underline{F(j)}}). \\nonumber\n\\end{eqnarray}\nTo simplify, all we need to consider is the intersection between $\\underline{F(j)}$ and the support of $(a_i^\\dag a_k + a_k^\\dag a_i)$. In this case the support of the excitation operator is $U_{ik} \\cup \\alpha_{ik} \\cup P_{ik}^0 \\cup \\{i,k\\}$. The form of the simplification will vary depending on these sets, but the process of reducing local operator products by exploiting the relationship between the three Pauli matrices is unchanged. In the cases when $i$ and $k$ are not both even, all that changes is the form of the excitation operator from Table~\\ref{tab2} that must be used.\n\n\\subsection{Double-excitation operators: $h_{ijkl} \\ (a^\\dag_i a^\\dag_j a_k a_l + a^\\dag_l a^\\dag_k a_j a_i)$}\nThe double-excitation operators involve four distinct indices, and are obviously the most algebraically complicated class of operators we are considering. The impractical number of sub-cases depending on the specific combination of indices $i,j,k,l$ means that we only outline the procedure for deriving algebraic expressions for this class of operators.\nThe fermionic commutation relations ensure that the following is true:\n\\begin{equation}\n(a^\\dag_i a^\\dag_j a_k a_l + a^\\dag_l a^\\dag_k a_j a_i) = (a^\\dag_i a_l) (a^\\dag_j a_k) + (a^\\dag_l a_i) (a^\\dag_k a_j).\n\\end{equation}\nAllowing for the integral $h_{ijkl}$ to be complex, we can write:\n\\begin{eqnarray}\nh_{ijkl} \\ (a^\\dag_i a^\\dag_j a_k a_l + a^\\dag_l a^\\dag_k a_j a_i) = [&\\Re\\{h_{ijkl}\\}(a^\\dag_i a_l a^\\dag_j a_k + a^\\dag_l a_i a^\\dag_k a_j) \\\\ + &\\Im\\{h_{ijkl}\\}\\ (a^\\dag_i a_l a^\\dag_j a_k - a^\\dag_l a_i a^\\dag_k a_j)]. \\nonumber\n\\end{eqnarray}\n\\noindent Since $(a^\\dag_i a_l a_j^\\dag a_k)^\\dag = a^\\dag_l a_i a^\\dag_k a_j$, we can simply consider the algebraic expression for the product of two operators of the form $a_i^\\dag a_j$ as given in Table~\\ref{tab2}, and then add or subtract it to its Hermitian conjugate. Each of the operators $a^\\dag_i a_l$ and $a_j^\\dag a_k$ will fit into one of the ten cases presented in Table~\\ref{tab2}. In multiplying out the algebraic expressions for these two products, what is important is the set \\{supp($a^\\dag_i a_l$) $\\cap$ supp($a_j^\\dag a_k$)\\}. Any qubits in this set will have a product of local operators acting on it which must be simplified. \n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nIndex parity & & Conditions & & Algebraic expression for $a_i^\\dag a_j$ \\\\\n\\hline\n & $i \\in P(j)$ & $j \\in U(i)$ & $|\\alpha_{ij}| $& \\\\\n\\hline\n\\multirow{1}{*}{$i,j$ even} & No & No & 1 & $\\frac{1}{4} X_{U_{ij} \\setminus \\alpha_{ij}}\\ Y_{\\alpha_{ij}}\\ Z_{P_{ij}^0 \\setminus \\alpha_{ij}} [Y_j X_i - X_j Y_i -i (X_j X_i + Y_j Y_i)]$ \\\\ \n\\hline\n\\multirow{3}{*}{$i$ odd, $j$ even}& No & No & 1 & $\\frac{1}{4} X_{U_{ij} \\setminus \\alpha_{ij}}\\ Y_{\\alpha_{ij}}\\ \\overline{Z}_{\\alpha_{ij}}\\ [(Y_j X_i - i X_j X_i)\\ Z_{P_{ij}^0} - (X_j Y_i + iY_j Y_i)\\ Z_{P_{ij}^2} ] $ \\\\\n& & & & \\\\\n & Yes & No & 0 & $\\frac{1}{4} X_{U_{ij}}\\ \\overline{Z}_i \\ [(Y_j Y_i - i X_j \\overline{X}_i Y_i)\\ Z_{P_{ij}^0} + (X_j X_i + iY_j X_i)\\ Z_{P_{ij}^2} ]$ \\\\ \n\\hline\n\\multirow{5}{*}{$i$ even, $j$ odd} & No & No & 1 & $\\frac{1}{4} X_{U_{ij} \\setminus \\alpha_{ij}}\\ Y_{\\alpha_{ij}}\\ \\overline{Z}_{\\alpha_{ij}}\\ [- (X_j Y_i + i X_j X_i)\\ Z_{P_{ij}^0} + (Y_j X_i - iY_j Y_i)\\ Z_{P_{ij}^1} ]$ \\\\\n& & & & \\\\\n & No & Yes & 1 & $\\frac{1}{4} X_{U_{ij} \\setminus j}\\ [-\\overline{X}_{\\alpha_{ij}}(Y_i - i X_i)\\ Y_{\\alpha_{ij}}\\ Z_{P_{ij}^0 \\setminus \\alpha_{ij}} + (i Y_i - X_i)\\ Z_{P_{ij}^1 \\cup j} ]$ \\\\\n & & & & \\\\\n & Yes & Yes & 0 & $\\frac{1}{4} X_{U_{ij} \\setminus j}\\ [ (X_i - i Y_i) + (i Y_i - X_i)\\ Z_{P_{ij}^1 \\cup j} ]$ \\\\ \\midrule\n\\hline\n\\multirow{7}{*}{$i,j$ odd} & No & No & 1 & $\\frac{1}{4} X_{U_{ij} \\setminus \\alpha_{ij}}\\ Y_{\\alpha_{ij}} \\overline{Z}_{\\alpha_{ij}}\\ [ -i X_j X_i Z_{P_{ij}^0} + Y_j X_i Z_{P_{ij}^1} - X_j Y_i Z_{P_{ij}^2} - i Y_j Y_i Z_{P_{ij}^3}]$ \\\\\n& & & & \\\\\n & Yes & No & 0 & $\\frac{1}{4} X_{U_{ij}}\\ \\overline{Z}_i [ (-i X_j Y_i Z_{P_{ij}^0} + Y_j Y_i Z_{P_{ij}^1}) + X_j X_i Z_{P_{ij}^2} + i Y_j X_i Z_{P_{ij}^3}]$ \\\\\n & & & & \\\\\n & No & Yes & 1 & $\\frac{1}{4} X_{U_{ij} \\setminus j}\\ [ -\\overline{X}_{\\alpha_{ij}}(Y_i Z_{P_{ij}^2} + i X_i Z_{P_{ij}^0}) Y_{\\alpha_{ij}} \\overline{Z}_{\\alpha_{ij}} - (X_i Z_{P_{ij}^1} - i Y_i Z_{P_{ij}^3}) Z_j ]$ \\\\\n & & & & \\\\\n & Yes & Yes &0 & $\\frac{1}{4} X_{U_{ij} \\setminus j}\\ [ \\overline{Z}_i (-i Y_i Z_{P_{ij}^0} + X_i Z_{P_{ij}^2}) + Z_j (-X_i Z_{P_{ij}^1} + i Y_i Z_{P_{ij}^3})]$ \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab2}The algebraic expressions for general products of the form $a_i^\\dag a_j$ in the Bravyi-Kitaev basis. These expressions vary in form depending on the parity of the indices $i$ and $j$, as well as on the overlaps between the parity and update sets of the indices. The notation $\\overline{O}_S$ is shorthand to indicate that the operator $O$ does {\\em not} operate on the qubits in the set $S$ (i.e. $Z_{P_{ij}^0} \\overline{Z}_j= Z_{P_{ij}^0 \\setminus j}$).}\n\\end{table}\n\n\\section{The molecular electronic Hamiltonian for the hydrogen molecule in the Bravyi-Kitaev basis}\\label{molham}\n\nThe molecular electronic Hamiltonian~(\\ref{molhameq}) may be divided into one and two-electron terms:\n\\begin{equation}\n\\hat{H}=\\sum_{i,j}h_{ij}a^\\dag_i a_j +\\frac{1}{2}\\sum_{i,j,k,l} h_{ijkl} a^\\dag_i a^\\dag_j a_k a_l = \\hat{H}^{(1)} + \\hat{H}^{(2)}.\n\\end{equation}\n\nWe treat molecular hydrogen in a minimal basis, so the sums above run over the four spin orbitals defined above. These spin orbitals will be indexed 0 through 3, as will be the fermionic creation and annihilation operators. We derive the simplified expressions for the individual terms of this Hamiltonian in the Bravyi-Kitaev basis. The overlap integrals $h_{ij}$ and $h_{ijkl}$ for $0 \\leq i \\leq 3$ are given in Table~\\ref{tab3}. These are the same as were used in \\cite{James} and were calculated using a restricted Hartree-Fock calculation in the PyQuante quantum chemistry package \\cite{Muller}. With these integrals and the algebraic expressions for second quantized operators given in Section~\\ref{pauli}, we can express the molecular electronic Hamiltonian for H$_2$ as a sum of products of Pauli matrices. In the next two subsections we consider the one- and two-electron Hamiltonians separately.\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{ |c| c| }\n\\hline\nIntegrals & Value (a.u.) \\\\\n\\hline\n$h_{00} = h_{11}$ & $-1.252477$ \\\\\n\\hline\n$h_{22} = h_{33}$& $-0.475934$\\\\\n\\hline\n$h_{0110} = h_{1001}$ & $\\ 0.674493$\\\\\n\\hline\n$h_{2332} = h_{3223}$ & $\\ 0.697397$\\\\\n\\hline\n$\\quad h_{0220} = h_{0330} = h_{1221} = h_{1331}$ &\\multirow{2}{*}{$0.663472$}\\\\\n$= h_{2002} = h_{3003} = h_{2112} = h_{3113}$ & \\\\\n\\hline\n$h_{0202} = h_{1313} = h_{2130} = h_{2310} = h_{0312} = h_{0132}$ & $0.181287$\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab3}The overlap integrals for molecular hydrogen in a minimal basis. The integrals were obtained through a restricted Hartree-Fock calculation in the PyQuante quantum chemistry package at an internuclear separation of $1.401000$ atomic units ($7.414 \\times 10^{-11}$ m).}\n\\end{center}\n\\end{table}\n\n\\subsection{The Bravyi-Kitaev Pauli representation of $\\hat{H}^{(1)}$}\\label{sevenone}\n\nWe can write the one-electron terms in the Hamiltonian as:\n\\begin{equation}\n\\hat{H}^{(1)} = h_{00}a_0^\\dag a_0 + h_{11}a_1^\\dag a_1 + h_{22}a_2^\\dag a_2 + h_{33}a_3^\\dag a_3.\n\\end{equation}\nUsing the expressions for number operators derived in Section~\\ref{bravkit}, we know that in the Bravyi-Kitaev basis, these operators are:\n\\begin{eqnarray}\n&a_0^\\dag a_0 = \\frac{1}{2}(\\mathbf{1} - \\sigma_0^z); \\\\\n&a_1^\\dag a_1 = \\frac{1}{2}(\\mathbf{1} - \\sigma_1^z \\sigma_0^z); \\\\\n&a_2^\\dag a_2 = \\frac{1}{2}(\\mathbf{1} - \\sigma_2^z); \\\\\n&a_3^\\dag a_3 = \\frac{1}{2}(\\mathbf{1} - \\sigma_3^z \\sigma_2^z \\sigma_1^z).\n\\end{eqnarray}\nWe now proceed to the simulation of $\\hat{H}^{(2)}$.\n\n\\subsection{The Bravyi-Kitaev Pauli representation of $\\hat{H}^{(2)}$}\n\nFollowing the work of Whitfield {\\it et al}.\\ \\cite{James}, $\\hat{H}^{(2)}$ simplifies to the following expression for molecular hydrogen in a minimal basis:\n\\begin{eqnarray}\n\\hat{H}^{(2)} = h_{0110}a_0^\\dag a_1^\\dag a_1 a_0 + h_{2332}a_2^\\dag a_3^\\dag a_3 a_2 &+ h_{0330}a_0^\\dag a_3^\\dag a_3 a_0 + h_{1221}a_1^\\dag a_2^\\dag a_2 a_1 \\\\ \n\\ + (h_{0220} - h_{0202})a_0^\\dag a_2^\\dag a_2 a_0 + (h_{1331} - h_{1313}) &a_1^\\dag a_3^\\dag a_3 a_1 + h_{0132}(a_0^\\dag a_1^\\dag a_3 a_2 + a_2^\\dag a_3^\\dag a_1 a_0) \\nonumber \\\\\n+ h_{0312}(a_0^\\dag a_3^\\dag a_1 a_2 &+ a_2^\\dag a_1^\\dag a_3 a_0). \\nonumber\n\\end{eqnarray}\nThis term in the Hamiltonian is made up of six Coulomb\/exchange operators and two double-excitation operators. Using Section~\\ref{pauli}, it is easy to give algebraic expressions for the Coulomb and exchange operators:\n\\begin{eqnarray}\n&a_0^\\dag a_1^\\dag a_1 a_0 = \\frac{1}{4}(\\mathbf{1} - \\sigma_0^z - \\sigma_1^z \\sigma_0^z + \\sigma_1^z); \\\\\n&a_2^\\dag a_3^\\dag a_3 a_2 = \\frac{1}{4}(\\mathbf{1} - \\sigma_2^z - \\sigma_3^z \\sigma_2^z \\sigma_1^z + \\sigma_3^z \\sigma_1^z); \\\\\n&a_0^\\dag a_3^\\dag a_3 a_0 = \\frac{1}{4}(\\mathbf{1} - \\sigma_0^z - \\sigma_3^z \\sigma_2^z \\sigma_1^z + \\sigma_3^z \\sigma_2^z \\sigma_1^z \\sigma_0^z ); \\\\\n&a_1^\\dag a_2^\\dag a_2 a_1 = \\frac{1}{4}(\\mathbf{1} - \\sigma_2^z - \\sigma_1^z \\sigma_0^z + \\sigma_2^z \\sigma_1^z \\sigma_0^z); \\\\\n&a_0^\\dag a_2^\\dag a_2 a_0 = \\frac{1}{4}(\\mathbf{1} - \\sigma_2^z - \\sigma_0^z + \\sigma_2^z \\sigma_0^z); \\\\\n&a_1^\\dag a_3^\\dag a_3 a_1 = \\frac{1}{4}(\\mathbf{1} - \\sigma_3^z \\sigma_2^z \\sigma_1^z - \\sigma_1^z \\sigma_0^z + \\sigma_3^z \\sigma_2^z \\sigma_0^z).\n\\end{eqnarray}\nThe two double-excitation operators are somewhat more complicated. As an example, we will derive the Pauli representation of $h_{0312}(a_0^\\dag a_3^\\dag a_1 a_2 + a_2^\\dag a_1^\\dag a_3 a_0)$. Following in Section~\\ref{pauli}, we consider $a_0^\\dag a_3^\\dag a_1 a_2$ as $(a_0^\\dag a_2) (a_3^\\dag a_1)$, a product of two operators of the form $a_i^\\dag a_j$. The term $a_0^\\dag a_2$ is of the type when $i$ and $j$ are both even, while the term $a_1^\\dag a_3$ is of the type when $i$ and $j$ are odd, and $i \\in P(j)$, $j \\in U(i)$, and $|\\alpha_{ij}| = 0$. Using the appropriate expressions from Table~\\ref{tab2}, we find the following:\n\\begin{eqnarray}\n& a_0^\\dag a_2 = \\frac{1}{4} (\\sigma^y_2 \\sigma^y_1 \\sigma^x_0 - \\sigma^x_2 \\sigma^y_1 \\sigma^y_0 - i \\sigma^x_2 \\sigma^y_1 \\sigma^x_0 -i \\sigma^y_2 \\sigma^y_1 \\sigma^y_0); \\\\\n& a_1^\\dag a_3 = \\frac{1}{4} (-i \\sigma^z_2 \\sigma^y_1 \\sigma^z_0 + \\sigma^z_2 \\sigma^x_1 - \\sigma^z_3 \\sigma^x_1 \\sigma^z_0 + i \\sigma^z_3 \\sigma^y_1).\n\\end{eqnarray}\n\n\\noindent Now we note that supp($a_0^\\dag a_2$) $\\cap$ supp($a_1^\\dag a_3 $) $= \\{2,1,0\\}$, and so we must expect to simplify local operator products on qubits with these indices. Taking the product, we find the following:\n\\begin{eqnarray}\na_0^\\dag a_2 a_1^\\dag a_3 = \\frac{1}{16} (&\\sigma^x_2 \\sigma^x_0 - i \\sigma^x_2 \\sigma^y_0 + \\sigma^x_2 \\sigma^z_1 \\sigma^x_0 - i \\sigma^x_2 \\sigma^z_1 \\sigma^y_0 \\\\\n + &i \\sigma^y_2 \\sigma^x_0 + \\sigma^y_2 \\sigma^y_0 + i \\sigma^y_2 \\sigma^z_1 \\sigma^x_0 + \\sigma^y_2 \\sigma^z_1 \\sigma^y_0 \\nonumber \\\\\n + &\\sigma^z_3 \\sigma^x_2 \\sigma^x_0 - i \\sigma^z_3 \\sigma^x_2 \\sigma^y_0 + \\sigma^z_3 \\sigma^x_2 \\sigma^z_1 \\sigma^x_0 - i \\sigma^z_3 \\sigma^x_2 \\sigma^z_1 \\sigma^y_0 \\nonumber \\\\\n + &i \\sigma^z_3 \\sigma^y_2 \\sigma^x_0 + \\sigma^z_3 \\sigma^y_2 \\sigma^y_0 + i \\sigma^z_3 \\sigma^y_2 \\sigma^z_1 \\sigma^x_0 + \\sigma^z_3 \\sigma^y_2 \\sigma^z_1 \\sigma^y_0). \\nonumber\n\\end{eqnarray}\n\nSince the integral $h_{0132}$ is real, we can simply add the above result to its Hermitian conjugate to find the expression for the double-excitation operator. Repeating the above procedure for the second double excitation operator, we arrive at the following results:\n\\begin{eqnarray}\na_0^\\dag a_3^\\dag a_1 a_2 + a_2^\\dag a_1^\\dag a_3 a_0 = \\frac{1}{8}(&- \\sigma_{2}^x\\sigma_{0}^x\\ + \\sigma_{2}^x\\sigma_{1}^z\\sigma_{0}^x\\ - \\sigma_{2}^y\\sigma_{0}^y\\ + \\sigma_{2}^y\\sigma_{1}^z\\sigma_{0}^y\\ - \\sigma_{3}^z\\sigma_{2}^x\\sigma_{0}^x\\ \\\\\n& + \\sigma_{3}^z\\sigma_{2}^x\\sigma_{1}^z\\sigma_{0}^x\\ - \\sigma_{3}^z\\sigma_{2}^y\\sigma_{0}^y\\ + \\sigma_{3}^z\\sigma_{2}^y\\sigma_{1}^z\\sigma_{0}^y); \\nonumber \\\\\na_0^\\dag a_1^\\dag a_3 a_2 + a_2^\\dag a_3^\\dag a_1 a_0 = \\frac{1}{8}(& \\sigma_{2}^x\\sigma_{0}^x\\ + \\sigma_{2}^x\\sigma_{1}^z\\sigma_{0}^x\\ + \\sigma_{2}^y\\sigma_{0}^y\\ + \\sigma_{2}^y\\sigma_{1}^z\\sigma_{0}^y\\ + \\sigma_{3}^z\\sigma_{2}^x\\sigma_{0}^x\\ \\\\\n& + \\sigma_{3}^z\\sigma_{2}^x\\sigma_{1}^z\\sigma_{0}^x\\ + \\sigma_{3}^z\\sigma_{2}^y\\sigma_{0}^y\\ + \\sigma_{3}^z\\sigma_{2}^y\\sigma_{1}^z\\sigma_{0}^y). \\nonumber\n\\end{eqnarray}\nThus, using the integrals from Table~\\ref{tab3} and the Pauli expressions for the number operators derived in Section~\\ref{sevenone}, as well as the Coulomb\/exchange operators and the double-excitation operators derived in this section, we can represent the molecular electronic Hamiltonian for the hydrogen molecule as a sum of products of Pauli matrices in the Bravyi-Kitaev basis:\n\\begin{eqnarray}\\label{svnnn}\n\\hat{H}_{BK} =\\ &-0.81261\\ \\mathbf{1}+0.171201\\ \\sigma_{0}^z\\ +0.16862325\\ \\sigma_{1}^z\\ -0.2227965\\ \\sigma_{2}^z\\ +0.171201\\ \\sigma_{1}^z \\sigma_{0}^z\\ \\nonumber \\\\\n&+0.12054625\\ \\sigma_{2}^z\\sigma_{0}^z\\ +0.17434925\\ \\sigma_{3}^z\\sigma_{1}^z\\ +0.04532175\\ \\sigma_{2}^x\\sigma_{1}^z\\sigma_{0}^x\\ +0.04532175\\ \\sigma_{2}^y\\sigma_{1}^z\\sigma_{0}^y\\ \\nonumber \\\\\n&+0.165868\\ \\sigma_{2}^z\\sigma_{1}^z\\sigma_{0}^z\\ +0.12054625\\ \\sigma_{3}^z\\sigma_{2}^z\\sigma_{0}^z\\ -0.2227965\\ \\sigma_{3}^z\\sigma_{2}^z\\sigma_{1}^z\\ \\nonumber \\\\\n&+0.04532175\\ \\sigma_{3}^z\\sigma_{2}^x\\sigma_{1}^z\\sigma_{0}^x\\ +0.04532175\\ \\sigma_{3}^z\\sigma_{2}^y\\sigma_{1}^z\\sigma_{0}^y\\ +0.165868\\ \\sigma_{3}^z\\sigma_{2}^z\\sigma_{1}^z\\sigma_{0}^z. \n\\end{eqnarray}\nThis Hamiltonian is isospectral to the Jordan-Wigner derived Hamiltonian \\cite{James}:\n\\begin{eqnarray}\\label{svntwn}\n\\hat{H}_{JW} =\\ &-0.81261\\ \\mathbf{1}+0.171201\\ \\sigma_{0}^z\\ +0.171201\\ \\sigma_{1}^z\\ -0.2227965\\ \\sigma_{2}^z\\ -0.2227965\\ \\sigma_{3}^z\\ \\nonumber \\\\\n&+0.16862325\\ \\sigma_{1}^z\\sigma_{0}^z\\ +0.12054625\\ \\sigma_{2}^z\\sigma_{0}^z\\ +0.165868\\ \\sigma_{2}^z\\sigma_{1}^z\\ +0.165868\\ \\sigma_{3}^z\\sigma_{0}^z\\ \\nonumber \\\\\n&+0.12054625\\ \\sigma_{3}^z\\sigma_{1}^z\\ +0.17434925\\ \\sigma_{3}^z\\sigma_{2}^z\\ -0.04532175\\ \\sigma_{3}^x\\sigma_{2}^x\\sigma_{1}^y\\sigma_{0}^y\\ \\nonumber \\\\\n&+0.04532175\\ \\sigma_{3}^x\\sigma_{2}^y\\sigma_{1}^y\\sigma_{0}^x\\ +0.04532175\\ \\sigma_{3}^y\\sigma_{2}^x\\sigma_{1}^x\\sigma_{0}^y\\ -0.04532175\\ \\sigma_{3}^y\\sigma_{2}^y\\sigma_{1}^x\\sigma_{0}^x.\n\\end{eqnarray}\nWriting the electronic Hamiltonians in the form of equations~(\\ref{svnnn}) and~(\\ref{svntwn}) allows for a comparison of the computational resources required to simulate them on a quantum computer. Not all tensor products of Pauli matrices that appear in these Hamiltonians commute with one another, so exponentiating them requires the use of a Trotter approximation. The next section details the Trotterization process for the Hamiltonian in the Bravyi-Kitaev basis.\n\n\\section{Trotterization}\\label{trott}\nIdeally, one could simulate the propagator $e^{-i \\hat{H} t}$, where $\\hat{H} = \\sum_{k} h_k$, by sequentially exponentiating the individual terms $h_k$ on a quantum simulator. However, $e^{-i \\hat{H} t} = \\prod{e^{-i h_k t}}$ only in the case that the set of $h_k$ all mutually commute. Both the Bravyi-Kitaev and Jordan-Wigner Hamiltonians contain terms that do not commute with one another, and so a Suzuki-Trotter approximation must be used. The first four orders of Suzuki-Trotter formulae are \\cite{QA}:\n\\begin{equation} e^{(A+B)t} \\approx (e^{At\/n}e^{Bt\/n})^n+O(t \\Delta t);\n\\end{equation}\n\\begin{equation} e^{(A+B)t} \\approx (e^{At\/2n}e^{Bt\/n}e^{At\/2n})^{n}+O(t (\\Delta t)^2);\n\\end{equation}\n\\begin{equation} e^{(A+B)t} \\approx (e^{\\frac{7}{24}At\/n}e^{\\frac{2}{3}Bt\/n}e^{\\frac{3}{4}At\/n}e^{\\frac{-2}{3}Bt\/n}e^{\\frac{-1}{24}At\/n}e^{Bt\/n})^{n}+O(t (\\Delta t)^3);\n\\end{equation}\n\\begin{equation} e^{(A+B)t} \\approx (\\prod{_{i=1}^5}e^{p_i At\/2n}e^{p_i Bt\/n}e^{p_i At\/2n})^n+O(t (\\Delta t)^4),\n\\end{equation}\n\n\\noindent where in the 4th order equation, the constants are given by:\n\\begin{equation}\np_1 =p_2 =p_4 =p_5 = \\frac{1}{4-4^{1\/3}}, \\qquad p_3=1-4p_1.\n\\end{equation}\n\n\\noindent The terms of both the Bravyi-Kitaev Hamiltonian and the Jordan-Wigner Hamiltonian can be broken into two subsets, where the terms in each subset all mutually commute but the subsets do not commute with one another. These groups are as follows:\n\\begin{eqnarray}\n\\hat{H}_{BK, Z} =\\ -&0.81261\\ \\mathbf{1}+0.171201\\ \\sigma_{0}^z\\ +0.16862325\\ \\sigma_{1}^z\\ -0.2227965\\ \\sigma_{2}^z\\ +0.171201\\ \\sigma_{1}^z \\sigma_{0}^z\\ \\nonumber \\\\\n+&0.12054625\\ \\sigma_{2}^z\\sigma_{0}^z\\ +0.17434925\\ \\sigma_{3}^z\\sigma_{1}^z\\ +0.165868\\ \\sigma_{2}^z\\sigma_{1}^z\\sigma_{0}^z\\ \\nonumber \\\\\n+&0.12054625\\ \\sigma_{3}^z\\sigma_{2}^z\\sigma_{0}^z\\ -0.2227965\\ \\sigma_{3}^z\\sigma_{2}^z\\sigma_{1}^z\\ +0.165868\\ \\sigma_{3}^z\\sigma_{2}^z\\sigma_{1}^z\\sigma_{0}^z; \\\\\n& \\nonumber \\\\\n\\hat{H}_{BK, XY} = \\ &0.04532175\\ \\sigma_{2}^x\\sigma_{1}^z\\sigma_{0}^x\\ +0.04532175\\ \\sigma_{2}^y\\sigma_{1}^z\\sigma_{0}^y\\ +0.04532175\\ \\sigma_{3}^z\\sigma_{2}^x\\sigma_{1}^z\\sigma_{0}^x\\ \\nonumber \\\\\n+&0.04532175\\ \\sigma_{3}^z\\sigma_{2}^y\\sigma_{1}^z\\sigma_{0}^y;\n\\end{eqnarray}\n\\begin{eqnarray}\n\\hat{H}_{JW, Z} =\\ -&0.81261\\ \\mathbf{1}+0.171201\\ \\sigma_{0}^z\\ +0.171201\\ \\sigma_{1}^z\\ -0.2227965\\ \\sigma_{2}^z\\ -0.2227965\\ \\sigma_{3}^z\\ \\nonumber \\\\\n+&0.16862325\\ \\sigma_{1}^z\\sigma_{0}^z\\ +0.12054625\\ \\sigma_{2}^z\\sigma_{0}^z\\ +0.165868\\ \\sigma_{2}^z\\sigma_{1}^z\\ +0.165868\\ \\sigma_{3}^z\\sigma_{0}^z\\ \\nonumber \\\\\n+&0.12054625\\ \\sigma_{3}^z\\sigma_{1}^z\\ +0.17434925\\ \\sigma_{3}^z\\sigma_{2}^z; \\\\\n& \\nonumber \\\\\n\\hat{H}_{JW, XY} = -&0.04532175\\ \\sigma_{3}^x\\sigma_{2}^x\\sigma_{1}^y\\sigma_{0}^y\\ +0.04532175\\ \\sigma_{3}^x\\sigma_{2}^y\\sigma_{1}^y\\sigma_{0}^x\\ +0.04532175\\ \\sigma_{3}^y\\sigma_{2}^x\\sigma_{1}^x\\sigma_{0}^y\\ \\nonumber \\\\\n-&0.04532175\\ \\sigma_{3}^y\\sigma_{2}^y\\sigma_{1}^x\\sigma_{0}^x.\n\\end{eqnarray}\nTo understand what computational resources are required for exponentiating operators of this kind, consider the example of the exponentiation of a fourfold product of $\\sigma^z$ matrices, $e^{i(\\sigma^z \\otimes \\sigma^z \\otimes \\sigma^z \\otimes \\sigma^z)}$, which is depicted in a circuit diagram in Figure~\\ref{zzzz}~\\cite{MikeIke}.\n\n\\begin{figure}[!h]\n\\centerline{ \\includegraphics[scale = .35]{figure3.png}}\n \\caption[Exponentiation of tensor products of Pauli-Z matrices]{A demonstration of how to exponentiate tensor products of Pauli matrices. First, the parity of the four qubits is computed with CNOT gates, and then a single-qubit phase rotation $R_z$ is applied. Then, we uncompute the parity with three further CNOT gates.\\label{zzzz}}\n\\end{figure}\n\nIn general, an $n$-fold tensor product of Pauli-Z matrices will require $2(n-1)$ CNOT gates and one single-qubit gate (SQG) to exponentiate on a quantum computer. If there are Pauli-X or -Y matrices in the tensor product, we must apply the single-qubit Hadamard or $R_x$ gate to change basis to the $X$ or $Y$ basis, respectively, before we compute the parity of the set of qubits with CNOT's, and also apply the inverse gates as part of the uncomputing stage \\cite{MikeIke}. These gates are given by:\n\\begin{equation}\nH = \\frac{1}{\\sqrt{2}} \\left[\\begin{array}{cc} 1&1\\\\1 & -1 \\end{array}\\right] \\quad \\quad \\quad R_x = \\frac{1}{\\sqrt{2}} \\left[ \\begin{array}{ll} 1&i\\\\i & 1 \\end{array}\\right] \n\\end{equation}\nThus, each non-$\\sigma^z$ term in a tensor product of Pauli matrices adds $2$ single-qubit gates to the cost of exponentiation. For example, the circuit for exponentiating the term $\\sigma_{3}^y\\sigma_{2}^x\\sigma_{1}^x\\sigma_{0}^y$ is depicted in Figure~\\ref{yxxy}\n\n\\begin{figure}[!h]\n\\centerline{ \\includegraphics[scale = .35]{figure4.png}}\n \\caption[\\label{yxxy} Exponentiation of tensor products of Pauli-X and -Y matrices]{A demonstration of how to exponentiate tensor products of Pauli-X and -Y matrices. First, the qubits are put in the correct basis by the application of $R_x$ or Hadamard gates. Then, the parity of the four qubits is computed with CNOT gates, and then a single-qubit phase rotation $R_z$ is applied. Then, we uncompute the parity with more CNOT gates, and finally change back to the computational (Z) basis.}\n \\label{yxxy}\n\\end{figure}\nUsing the resource counting methods detailed above, we can count the number of single-qubit gates (SQG's) and CNOT gates required to exponentiate (for arbitrary propagation time) the subsets of the Hamiltonians for both encodings. The results of this analysis are in Table~\\ref{tab4}.\n\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{| c |c| c | c|}\n\\hline\n & SQG's & CNOT's & Totals \\\\\n\\hline\n$\\hat{H}_{BK,Z}$ & 10 & 24 & 34 \\\\\n\\hline\n$\\hat{H}_{BK,XY}$ & 20 & 20 & 40 \\\\\n\\hline\nTotals & 30 & 44 & {\\bf 74}\\\\\n\\hline\n\\hline\n$\\hat{H}_{JW,Z}$ & 10 & 12 & 22 \\\\\n\\hline\n$\\hat{H}_{JW,XY}$ & 36 & 24 & 60 \\\\\n\\hline\nTotals & 46 & 36 & {\\bf 82} \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab4}The number of single-qubit gates and CNOT gates required to exponentiate subsets of the electronic Hamiltonian for the hydrogen molecule, represented in terms of spin variables through either the Bravyi-Kitaev transformation or the Jordan-Wigner transformation.}\n\\end{center}\n\\end{table}\n\nWe now have the tools to compare the number of gates required to compute the ground state eigenvalue of either the Bravyi-Kitaev Hamiltonian or the Jordan-Wigner Hamiltonian to chemical precision ($\\pm 10^{-4}$ a.u). Due to the small size of our model of the hydrogen system, it is easy for a classical computer to simulate the behavior of the quantum simulator. The true propagator $U = e^{-i \\hat{H} t}$ can be computed to sufficient precision by a matrix exponential function in Mathematica or a similar software package. Time evolution of the ground state by the true propagator will result in phase evolution:\n\\begin{equation}\nU \\ket{\\psi_g} = e^{-i E_g t} \\ket{\\psi_g}.\n\\end{equation}\nWe can therefore compute the exact eigenvalue as follows:\n\\begin{equation}\n\\bra{\\psi_g} U \\ket{\\psi_g} = \\bra{\\psi_g} e^{-i E_g t} \\ket{\\psi_g} = e^{-i E_g t}.\n\\end{equation}\nWe set the propagation time to unity, and extract the true eigenvalue $E_g$ from the complex phase $e^{-i E_g}$. To approximate the eigenvalue, we use a Suzuki-Trotter approximation to the true propagator, $\\tilde{U}$, and perform an analogous procedure:\n\\begin{equation}\n\\frac{\\bra{\\psi_g} \\tilde{U} \\ket{\\psi_g}}{|\\bra{\\psi_g} \\tilde{U} \\ket{\\psi_g}|} = e^{-i \\tilde{E}_g t}.\n\\end{equation}\nThe approximation to the true ground state eigenvalue, $\\tilde{E}_g$, becomes better as we increase the number of Trotter steps $n$. Figure~\\ref{energy} below plots the estimated eigenvalues of the minimal basis Jordan-Wigner and Bravyi-Kitaev Hamiltonians as a function of the number of gates required, for the first four orders of Suzuki-Trotter formulae.\n\nWe now compare this result to previous estimates. The benchmark is the gate count given in \\cite{James} for approximating the Jordan-Wigner Hamiltonian's ground state eigenvalue. It is clear from Figure~\\ref{energy} that our first order approximation requires $\\approx900$ gates to obtain chemical precision for the Jordan-Wigner Hamiltonian, while the gate estimate in \\cite{James} was about $500$ for the same task. This discrepancy arises from the fact that any number of variants on the first order Suzuki-Trotter formula could have been used in \\cite{James}. Given a noncommuting set of Hamiltonian terms, there is some optimal ordering that will produce the best accuracy. It is not possible to know in advance which ordering is optimal, and given that the number of terms in an electronic Hamiltonian scales as $O(n^4)$, in general it is difficult to optimize over the space of possible orderings. We have used the most na\\\"{i}ve variant of the first order Suzuki-Trotter formula in Figure~\\ref{energy}:\n\\begin{equation}\ne^{-i \\hat{H} t} = e^{-i(\\hat{H}_Z + \\hat{H}_{XY}) t} \\approx (e^{-i \\hat{H}_Z \\frac{t}{n}} e^{-i \\hat{H}_{XY} \\frac{t}{n}})^n.\n\\end{equation}\n\n\\begin{figure}[!h]\n\\centerline{ \\includegraphics[scale = .6]{figure5.png}}\n \\caption[Eigenvalue approximation]{\\label{energy}The approximation to the ground state eigenvalue, for both the Bravyi-Kitaev Hamiltonian (squares) and Jordan-Wigner Hamiltonian (circles), as a function of the number of gates required. The solid curves are the first order Suzuki-Trotter approximations, the dot-dashed second order, the dotted third order, and the dashed fourth. The dotted horizontal line represents the true eigenvalue, while the solid lines above and below represent the bounds for chemical precision.}\n\\end{figure}\nHowever, due to the small size of our model of the hydrogen molecule, it is easy to find an ordering that produces better accuracy. A second, more sophisticated, variant of the first order formula is to arrange the terms in $\\hat{H}_{Z}$ and $\\hat{H}_{XY}$ in order of descending coefficient magnitude. For example, for the Bravyi-Kitaev Hamiltonian, we have:\n\\begin{eqnarray}\n&\\hat{H}_{Z}: \\{h_{Z0}, h_{Z1}, h_{Z2}, \\dots \\} = \\{ -0.81261\\ \\mathbf{1} , -0.2227965\\ \\sigma_{2}^z , -0.2227965\\ \\sigma_{3}^z\\sigma_{2}^z\\sigma_{1}^z , \\dots \\}; \\\\\n&\\hat{H}_{XY}: \\{h_{XY0}, h_{XY1}, h_{XY2}, \\dots \\} = \\{ 0.04532175\\ \\sigma_{2}^x\\sigma_{1}^z\\sigma_{0}^x, 0.04532175\\ \\sigma_{2}^y\\sigma_{1}^z\\sigma_{0}^y, \\dots \\}.\n\\end{eqnarray}\nThen, we approximate the propagator by alternately exponentiating one term from the ordered list of $\\hat{H}_{Z}$ terms and one term from the ordered list of $\\hat{H}_{XY}$ terms until we have used all terms from $\\hat{H}_{XY}$. Then we exponentiate the rest of $\\hat{H}_{Z}$:\n\\begin{equation}\ne^{-i \\hat{H} t} \\approx (e^{-i h_{Z0} \\frac{t}{n}} e^{-i h_{XY0} \\frac{t}{n}} e^{-i h_{Z1} \\frac{t}{n}} e^{-i h_{XY1} \\frac{t}{n}} \\cdots e^{-i h_{XY3} \\frac{t}{n}} e^{-i h_{Z4} \\frac{t}{n}} e^{-i h_{Z5} \\frac{t}{n}} \\cdots)^n.\n\\end{equation}\nWith this method, we find that the number of gates required to obtain a chemical precision estimate of the ground state eigenvalue of the Jordan-Wigner Hamiltonian is $\\approx 300$, fewer than the result from \\cite{James}. Figure~\\ref{approx} compares the eigenvalue approximations for the na\\\"{i}ve first order method and the more sophisticated variant.\n\n\n\\begin{figure}[!h]\n\\centerline{ \\includegraphics[scale = .6]{figure6.png}}\n \\caption[Eigenvalue approximation]{\\label{approx}The approximation to the ground state eigenvalue, for both the Bravyi-Kitaev Hamiltonian (squares) and Jordan-Wigner Hamiltonian (circles), as a function of the number of gates required. The solid curve is the na\\\"{i}ve first order Suzuki-Trotter approximation, while the dashed curve is the result from alternating the noncommuting terms. The dotted horizontal line represents the true eigenvalue, while the solid lines above and below represent the bounds for chemical precision. The ground state eigenvalue of the Bravyi-Kitaev Hamiltonian can be approximated to chemical precision with 222 gates, while it takes 328 gates to do the same for the Jordan-Wigner Hamiltonian.}\n\\end{figure}\n\nThe point is that the systematic advantage of the Bravyi-Kitaev method over the Jordan-Wigner method is not obscured by the kind of term-ordering optimization that we have demonstrated above. Exponentiating the Bravyi-Kitaev Hamiltonian requires $74$ gates per first order Trotter step (of any variant), while the Jordan-Wigner Hamiltonian requires $82$ gates per first order Trotter step. To obtain a precision of $\\pm 10^{-4}$ a.u to the true eigenvalue with the na\\\"{i}ve first order Suzuki-Trotter approximation requires $11$ Trotter steps for both the Bravyi-Kitaev and Jordan-Wigner Hamiltonian, for a total cost of $814$ gates versus $902$ gates. With the noncommuting terms intermixed, it takes only $3$ Trotter steps to obtain the same precision for the Bravyi-Kitaev Hamiltonian, and 4 Trotter steps for the Jordan-Wigner Hamiltonian. Thus, if we intermix the noncommuting terms, the Bravyi-Kitaev transformation allows one to utilize $222$ gates instead of the $328$ gates required by the Jordan-Wigner transformation to obtain an equally precise estimate of the hydrogen molecule's ground state eigenvalue when using a first order Suzuki-Trotter approximation. When using higher-order Suzuki-Trotter approximations to obtain better than chemical precision, the gate savings increases (Fig.~\\ref{savings}).\n\n\\begin{figure}[!h]\n\\centerline{ \\includegraphics[scale = .6]{figure7.png}}\n \\caption[Eigenvalue approximation]{\\label{savings}The gate savings of using the Bravyi-Kitaev method instead of the Jordan-Wigner method, as a function of the precision in the estimate of the ground state eigenvalue for the first four orders of Suzuki-Trotter formulae. The vertical line is the threshold error for chemical precision. The triangle data points are first order, the squares second, the circles third, and the diamonds fourth.}\n\\end{figure}\n\n\n\\section{Conclusions}\\label{conc}\n\nIn this paper we have worked out a detailed application of the Bravyi-Kitaev transformation to Hermitian second quantized operators that appear in quantum chemical Hamiltonians. We suggest that this transformation should replace the Jordan-Wigner transformation for fermionic quantum simulation algorithms. We have demonstrated that the Bravyi-Kitaev transformation results in a small reduction in the number of gates, from $328$ gates to $222$ gates, required to implement a quantum simulation algorithm for electron dynamics in the simplest possible molecular system of H$_2$ in a minimal basis.\n\nIn some sense, molecular hydrogen in a minimal basis is a poor showcase of the power of the Bravyi-Kitaev transformation. Our description of this molecule utilizes four molecular orbitals, and hence four qubits. The spin Hamiltonians we derive using either the Bravyi-Kitaev transformation or the Jordan-Wigner Hamiltonian involve four-local Pauli tensor products, the result being that the cost of simulating time evolution under the Bravyi-Kitaev Hamiltonian on a quantum computer is only slightly reduced from that for the Jordan-Wigner Hamiltonian. However, were we to use a more sophisticated description of the H$_2$ --- for example, with eight molecular orbitals --- the Jordan-Wigner spin Hamiltonian would contain up to eight-local Pauli tensor products, while the Bravyi-Kitaev spin Hamiltonian would not. Given the asymptotically better $O(\\log n)$ scaling of the Bravyi-Kitaev method as compared to the $O(n)$ scaling of the Jordan-Wigner transformation, the difference between the two methods will become greater for larger basis sets and larger molecules --- the simulation of which is, after all, is the true goal of quantum simulation for quantum chemistry, since the small molecules are within the reach of conventional computers. However, by showing that the Bravyi-Kitaev method is more efficient for the smallest conceivable chemical system, we have demonstrated that there is no algorithmic overhead inherent to the Bravyi-Kitaev method that must be overcome by scaling up the size of problems to which it is applied. We have demonstrated the superior efficiency of the Bravyi-Kitaev transformation for all quantum chemical simulations. Thus, making use of the Bravyi-Kitaev transformation for fermionic quantum simulation will make simulations of larger molecules and with larger basis sets more readily accessible to experiment.\n\n\n \\section{Acknowledgments}\nThe authors thank the Aspuru-Guzik group for their hospitality during the summers of 2011 and 2012, when parts of this work were completed. We are indebted to Jarod Maclean, John Parkhill, Sam Rodriques, Joshua Schrier, Robert Seeley, and James Whitfield for productive discussions. This project is supported by NSF CCI center, ``Quantum Information for Quantum Chemistry (QIQC)\", award number CHE-1037992, by NSF award PHY-0955518 and by AFOSR award no FA9550-12-1-0046.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\nThe technique of the $Y$-formalism developed in a series of papers \n\\cite{Y-formalism:2001ab}-\\cite{Y-formalism:2007ab} has been used to derive \nthe complete \noperator product expansions for the composite operators involved in the \npure spinor string theory formalism \\cite{Berkovits:2000fe}. \nThe $Y$-formalism is very useful to compute the contact terms and the anomaly terms in the OPE's. \nIt is based on the observation that, in general to derive those terms only the local structure of the theory is needed and not the global information on the space. Thus, for instance, given a system whose \nnon-trivial information is encoded into an algebraic curve (such as $\\beta-\\gamma$ system \non a hypersurface, or a topological string on a Calabi-Yau space), one can regard the system as if \nit would be free using the $Y$-formalism to impose that the fields are constrained to live on a hypersurface. Before discussing the details of the formalism and some applications to $\\beta-\\gamma$ system \\cite{Witten:2005px}-\\cite{Tan:2006by}, we would like to present some interesting models to be analyzed using the present technique. \n\nWe start with some interesting 4d models. We consider their relations \nwith complex algebras generated by the currents. \n\nThe first model is described by four \ncoordinates $\\gamma^i$ (with $i=1,\\dots,4$) living on the complex plane ${\\bf C}^4$ (no constraint for the moment). In terms of them, one can construct 16 currents that form the $Sl(4)\\oplus Gl(1)$ algebra \nwith generators \n\\begin{equation}\n\\label{algebrafour}{\nJ_{i}^{~j} = \\beta_i \\gamma^j - \\frac{1}{ 4}\\delta_i^{~j} J \\,, ~~~~~\nJ = \\beta_i \\gamma^i\\,,\n}\n\\end{equation}\nwhere we have introduced the conjugate momentum $\\beta_i$ for each coordinate $\\gamma^i$. \nThe level of the algebra is easily computable by computing the double poles of the generators \nand one easily gets \n\\begin{equation}\\label{doublepoles}\nJ_{i}^{~j}(z) J_{k}^{~l}(w) \\rightarrow - \\frac{\\delta_i^l \\delta_k^j }{ (z-w)^2} + {\\rm first~order~poles}\\,, ~~~~~\nJ(z) J(w) \\rightarrow - \\frac{4 }{ (z-w)^2},\n\\end{equation}\nand therefore we have an affine algebra of the form $Sl(4)_{-1} \\oplus Gl(1)_{-4}$. \nSo, the character \n\\begin{equation}\n\\label{cfour}{\nZ_{{\\bf C}^4}(t_x,t_y,t_z,t_t|q) =\n\\frac{1}{\\sqrt{t_x t_y t_z t_t}}\n\\frac{\\eta^4(q) }{ \\theta_1(t_x|q) \\theta_1(t_y|q)\\theta_1(t_t|q)\\theta_1(t_ztq) }\n},\n\\end{equation}\ncan be expanded into a sum of characters of the two algebras \n\\begin{equation}\n\\label{deco}{\nZ_{{\\bf C}^4}(t_x,t_y,t_z,t_t|q) = \\sum_{K} \\chi_K^{SL(4), k=-1}(t_1, t_2, t_3|q) f_K(t)\\eta_K(q)\n},\n\\end{equation}\nwhere the sum over $K$-representations of $SL(4)$ and $f_K(t)$ are functions of $t= (t_x t_y t_z t_t)^{1\/4}$. The parameters $t_i$ are associated to the diagonal generators of $SL(4)$. \n\nThe second example is constructed by imposing an additional constraint. \nOn the set of coordinates $\\gamma^i$ we impose the constraint \n\\begin{equation}\\label{conio}\n\\gamma^1 \\gamma^2 - \\gamma^3 \\gamma^4 =0\\,.\n\\end{equation}\nThis is the well-known conifold, which is a singular Calabi-Yau space and it is a non-compact \ntoric variety with three dimensional complex directions. It is denoted by ${{\\bf C}^4[\\gamma^i]\/ \n\\langle \\gamma^1 \\gamma^2 - \\gamma^3 \\gamma^4=0 \\rangle}$ in algebraic topology and \nas a toric variety is denoted by $C^4\/C^*$. \nIn this case there are only 7 currents preserving the constraint, \nby denoting $g_{ij}$ the metric in ${\\bf C}^4$ with \nnon-trivial entries $g_{12} = - g_{34} =1$, we have \n\\begin{equation}\n\\label{currentsconi}{\nJ_{[ij]} = \\beta_{[i} g_{j]k} \\gamma^k\\,, ~~~~~~ J = \\beta_i \\gamma^i\\,.\n}\n\\end{equation}\nThe third model is a deformation of the previous one, by adding a small deformation to the constraint \n$$\n\\gamma^1 \\gamma^2 - \\gamma^3 \\gamma^4 =0 \\rightarrow \n\\gamma^1 \\gamma^2 - \\gamma^3 \\gamma^4 =\\epsilon \\,.\n$$\nIn this case the constraint is invariant under only the rotations of $SO(4)$ and it is not \nhomogeneous so the currents $J$ do not generate a symmetry of the model. \n\nIn the same way one can define the projective space by gauging the scale symmetry. For \nexample the projective space ${\\bf P}^3$ can be described in terms of the currents\n\n\\begin{equation}\n\\label{algebra}{\nJ_{i}^{~j} = w_i x^j - \\frac{1}{ 4}\\delta_i^{~j} J \\,, ~~~~~\nJ = w_i x^i\\,,\n}\n\\end{equation}\nwhere $J_i^{~j}$ generate $SL(4)$ and $J$ generates $GL(1)$. \nIn the following we will discuss the details and the application of the $Y$-formalism to the case of projective \nspaces. They can be treated as gauged linear sigma models, but there are interesting anomalies \nthat have to be taken into account. In addition, the projective spaces are useful to define Calabi-Yau space \nby selecting suitable sections in the space itself. \n\nThere are also some interesting 2d models that can be considered. We can again start with the obvious case of two free coordinates\n\\begin{equation}\n\\label{fourmodels}{\n{\\bf C}^2 \\rightarrow SL(2)_{-1} \\otimes GL(1)_{-2}\\,.\n}\n\\end{equation}\nThen, we can impose a simple constraint \n$$\n{{\\bf C}^2[x,y] \\over \\langle xy = 0 \\rangle} \\rightarrow SO(2)_{-1} \\otimes GL(1)_{+2}\\,, \n$$\nand finally, we can deform it to get \n$$\n{\\bf C}^* = GL(1) = {{\\bf C}^2[x,y] \\over \\langle xy =1 \\rangle} \\rightarrow SO(2)_{-1} \\,,\n$$\n$$\n{\\bf P}^1 = \\frac{{\\bf C}^2 }{ {\\bf C}^*} \\rightarrow SL(2)_{-2}\n~~~~~~{\\rm charges~of}~[x,y]=(1,1).\n$$\nIn all cases, we can use the $Y$-formalism to study the OPE's of the \ngauge-invariant operators without \nsolving the constraint explicitly. \n\nThe analysis of the $\\beta-\\gamma$ systems on hypersurfaces has been discussed in some papers \nuntil now. We would like to mention the pioneering work \\cite{Malikov:1998dw}-\\cite{Gorbounov:2000ac} on the construction \nof chiral vertex algbebras. These constructions already encode some of the conformal field theory \nanalysis for $\\beta-\\gamma$ systems. However, they never explicitly compute the CFT algebra in presence of hypersurface constraints except some simple cases when the constraints are \nsolvable in a simple way. Nevertheless their analysis was pivotal for more recent \ndevelopments \\cite{Witten:2005px,Nekrasov:2005wg}. \n\nThe main issue is the quantization of a system in presence of constraints. This is a well-known \nproblem in quantum field theory and it has been discussed in the literature since the \nadvent of quantum mechanics and quantum field theory. Nonetheless, in the case of 2d field \ntheories such as string theories and sigma models, one has the advantage of performing the \ncomputations in an explicit and exact way using the radial quantization technique and using the \nconformal field theory methods. In this regard, one would like to maintain such a strong feature even \nin presence of constraints. \nTherefore, one has to treat the constraints in a radically different way by imposing them at each step of computation without actually solving them. One example is the computation of the propagator \nbetween constrained fields. One can proceed as follows: first compute the correlator as it would have been free, then modify it consistently with the contraints. This procedure \nis encoded in the $Y$-formalism and it gives a systematical way to compute the correlation \nfunctions among different CFT operators. It can be used \nto compute all possible OPE's among gauge-invariant currents (which are \nnot sensible to the details of the \"gauge-fixing\" procedure). In addition, \nin order to give consistent results one has to check if all $Y$-dependent terms \ndrop out from the computation of gauge-invariant quantities and this provides a strong check \non the formalism itself. \n\nIn this article, we develop the $Y$ formalism for computation of generic $\\beta-\\gamma$ systems. It is pointed out that \nat the present we are not able to formulate the $Y$-formalism in general and some additional ingredient might have been introduced. However, there are\na vast number of applications which can be done even by using the present status of the formalism. \n\nThe paper is organized as follows: in sec. 2 we briefly review the curved $\\beta-\\gamma$ systems \nand we give some notations. In sec. 3, we develop the $Y$-formalism for $\\beta-\\gamma$ systemss for general systems first and then for quadratic and partly quadratic constraints. We point \nout some obstructions of the formalism. In sec. 4, we consider the gauge linear sigma model \ncounterparts of some examples. In sec. 5, we discuss additional variables and we make \nthe contact with the pure spinor formalism. In sec. 6, we generalize the $Y$-formalism to \nsuperprojective spaces. \n\n\\section{Curved $\\beta-\\gamma$ systems}\n\nA non-linear $\\beta-\\gamma$ system is specified by a map \n\\begin{equation}\n\\gamma: \\Sigma \\rightarrow X,\n\\end{equation}\nand a $(1, 0)$ form $\\beta$ on $\\Sigma$, valued in \nthe pull-back $\\gamma^{*}(T^{*}X)$, where $\\Sigma$ is the world-sheet Riemann surface\nand $X$ is a complex target space manifold. If $\\lbrace U_{(\\alpha)}\\rbrace$ is an open covering of $X$ and $\\gamma^{i}$ are local coordinates in $U_{(\\alpha)}$,\nthe $\\beta-\\gamma$ system is described by the action \n\\begin {equation}\\label{action}\n S = \\int \\beta_{i}\\bar\\partial\\gamma^{i},\n\\end{equation}\nwhich becomes locally linear and free in $U_{(\\alpha)}$.\nThe basic OPE is \n\\begin{equation}\n\\beta_{i}(y)\\gamma^{j}(z) = {{\\delta_{i}^{\\enskip j}}\\over{y - z}}.\n\\end{equation}\nThe action is invariant under diffeomorphisms\n\\begin{equation}\\label{diff0}\n\\delta \\gamma^i = V^i(\\gamma)\\,, ~~~~\n\\delta \\beta_i = - \\beta_j \\partial_i V^j(\\gamma)\\,. \n\\end{equation}\nwhere $V = V^{i}\\partial_{i}$ is a holomorphic vector field on $X$ and we use \nthe notation\n$\\partial_{i} = {{\\partial} \\over {\\partial \\gamma^{i}}}$. \nThe corresponding current is \n\\begin{equation}\\label{J}\nJ_{V} = \\beta_{i}V^{i}.\n\\end{equation}\nGiven two holomorphic vector fields $V$ and $U$, we can compute the OPE\n\\begin{equation}\nJ_{V}(y) J_{U}(z) = - {{\\partial_{i}V^{j}\\partial_{j}U^{i}(z)}\\over{(y-z)^{2}}} \n+ {{J_{[V,U]}}\\over{y-z}} - {{(\\partial_{k}\\partial_{i}V^{j})\n(\\partial_{j}U^{i})}\\over{y-z}} \\partial \\gamma^k.\n\\end{equation}\nThe last term shows a failure of closure for diffeomorphisms and it is a \npossible source of an anomaly that arises if this term cannot be reabsorbed \nby a redefinition of the \ncurrents. As shown in \\cite{Witten:2005px, Nekrasov:2005wg}, this obstruction arises if the \nfirst Pontryagin class $p_{1}(X)$ of $X$ does not vanish. In fact the\ndiffeomorphisms must be used to glue together the different patches $U_\n{(\\alpha)} $ of $X$ and the anomaly, which arises if $p_{1}(X)\\ne 0$, is \nindeed an obstruction for $X$ to be defined globally.\nAnother anomaly is present \\cite{Witten:2005px, Nekrasov:2005wg} if the product $ c_{1}(\\Sigma)c_\n{1}(X)$ of the first Chern classes of $\\Sigma$ and $X$ does not vanish.\nHowever, this anomaly does not appear if one works on a world-sheet with\n $ c_{1}(\\Sigma) = 0 $. \nHaving done these dutiful specifications, we will be no more concerned with \nthese obstructions henceforth.\\par\n \n\nIn a vast class of interesting models the target space \nmanifold $X$ is a hypersurface in $n$ dimensions defined by one or more \nconstraints. A well-known example is that of pure spinors in $D=10$, where \nthe manifold is a cone over $ SO(10)\/U(5)$ \\cite{Berkovits:2005hy}.\nIn this paper we will treat mainly the case of only one \nconstraint\n\\begin{equation} \\label{constraint}\n\\Phi(\\gamma) = \\sum_{h=0}^{n}{ \\Phi^{(h)}(\\gamma)} =0\\,,\n\\end{equation}\nwith $\\Phi^{(h)}$ being a homogeneous function of degree $h$.\nIn presence of the constraint (\\ref{constraint}), the action\n(\\ref{action}) is invariant under the gauge symmetry \n\\begin{equation}\\label{gaugesymm}\n\\delta \\beta_i = \\lambda \\partial_{i} \\Phi(\\gamma)\\,.\n\\end{equation} \nNotice that the gauge parameter $\\lambda $ is a $(1, 0)$ form.\nThis setting is \nuseful to study the example of hypersurfaces in projective spaces.\nThe constraint $\\Phi(\\gamma)=0$ to be embedded into a projective space needs \nto be homogeneous of some degree $h > 1$ with respect to the rescaling of the \ncoordinates so that only a term survives in (\\ref{constraint}) and not only \nthe action but also the constraint are invariant under the scale transformation \n$\\gamma^i \\rightarrow \\Lambda \\gamma^i$ where $\\Lambda$ is a scale. \nIf this symmetry is gauged, so that $\\Lambda$ becomes local, the model describes a projective space. \nTherefore, for a hypersurface in a projective space there are two sets of \ngauge symmetries and of constraints. The first set is composed by a \nscaling gauge symmetry and a linear constraint $\\beta_\n{i}\\gamma^{i} = 0 $.\nThe second one is determined by the constraint \n$\\Phi(\\gamma)=0$ which is non-linear and by the gauge symmetry \n(\\ref{gaugesymm}) which is non-linear. \n\n\n\\section{$Y$-formalism for constrained $\\beta-\\gamma$ systems}\n\\subsection{$Y$-formalism}\n\nThe usual approach to compute OPE's for a constrained $\\beta-\\gamma$ \nmodel consists in solving first the constraints in a given chart where the\nmodel becomes free, computing the free OPE's for this reduced system and then\nreconstructing the OPE's for the original operators. \n\nThe $Y$-formalism avoids this procedure by postulating the basic OPE among \n$\\beta_{i}$ and $\\gamma^{j}$ in the form\n\\begin{equation} \\label{basic}\n< \\beta_{i}(y)\\gamma^{j}(z)> = {{1}\\over{y-z}} (\\delta_{i}^{\\enskip j} - \nK_{i}^{\\enskip j}(z)),\n\\end{equation}\nand choosing $K_{i}^{\\enskip j} $ by demanding that the OPE's of $\\beta_i$\n(and therefore of any operator) with the constraints vanish. \nNote that $K_{i}^{\\enskip j} $ depends on a non-covariant field $Y_i$. \nFor instance, \nin the case of pure spinors \nwhere the constraint is $\\lambda \\Gamma^{m} \\lambda = 0$, the basic OPE is\n\\begin{equation}\n<\\omega_{\\alpha}(y)\\lambda^{\\beta}(z)> = {{1}\\over{y-z}} \n(\\delta_{\\alpha}^{\\enskip \\beta} - \\frac{1}{2} (\\Gamma_{m}\\lambda)_{\\alpha}\n(Y\\Gamma^{m})^{\\beta}(z)),\n\\end{equation}\nwith $Y_{\\alpha} \\equiv {v_{\\alpha} \\over (v\\lambda)}$, $v_{\\alpha}$ being a \nconstant spinor. This basic OPE was proposed in \\cite{Berkovits:2000fe} \nfor the first\ntime. If one makes use of it to compute OPE's among gauge-invariant operators,\n one obtains poles with $Y$-dependent contributions. \nHowever, it has been shown in \\cite{Y-formalism:2005ac} that if one introduces\n suitable $Y$-dependent corrections in the definition of these operators, \none can obtain $Y$-independent OPE's and they coincide with\nthe OPE's obtained by the usual method mentioned at the beginning of this \nsection.\n\nIn this section we will extend the $Y$-formalism to $\\beta-\\gamma$ system\ndescribed by the action (\\ref{action}) on a hypersurface defined by the\nconstraint (\\ref{constraint}). In particular, we shall discuss when this\nformalism is applicable and what its limitations are.\n\nLet us consider a constrained $\\beta-\\gamma$ model\n\\begin{equation}\\label{action0}\nS = \\int \\beta_{i}\\bar\\partial\\gamma^{i},\n\\end{equation}\nwhere the index $i$ runs over $1, \\cdots, N$ and this system is\ncharacterized by the constraint \n\\begin{equation} \\label{constraint2}\n\\Phi(\\gamma) = \\sum_{h=0}^{n}{ \\Phi^{(h)}(\\gamma)} = 0. \n\\end{equation}\nHere \n\\begin{equation}\n\\Phi^{(h)}(\\gamma) = {{1}\\over{h}} g_{i_{1} \\cdots i_{h}}\\gamma^{i_{1}} \n\\cdots \\gamma^{i_{h}},\n\\end{equation}\nis homogeneous in $\\gamma$ of degree $h$. \nLet us introduce the following notation: \n\\begin{equation}\n\\Phi_{i_{1} \\cdots i_{p}}(\\gamma) =\n\\partial_ {i_{1}} \\cdots \\partial_{i_p} \\Phi(\\gamma),\n\\end{equation}\nwith the definition of \n$\\partial_{i} \\equiv {{\\partial}\\over{\\partial \\gamma^{i}}}$.\nThe action (\\ref{action0}) has a local symmetry \n\\begin{equation}\n\\delta \\beta_{i} = \\lambda \\Phi_{i},\n\\end{equation}\nwhere $\\lambda$ is a local gauge parameter.\nThe classically gauge-invariant operators are \n\\begin{equation}\nT^0 = \\beta_i \\partial \\gamma^i,\n\\end{equation}\nand\n\\begin{equation}\nJ^0_{rs} = \\beta_{[r} \\Phi_{s]}.\n\\end{equation}\nThese currents are conserved since they leave invariant the constraint and \nthe action. \n\nThe other gauge-invariant and conserved currents can exist depending on \nthe form of $\\Phi(\\gamma)$. For instance let us consider a group $G$ of rigid transformations: \n\\begin{equation}\n\\delta \\gamma^i = \\delta\\Lambda^I (P_I)^i_{\\enskip j}\\gamma^{j}\\,, ~~~~\n\\delta \\beta_i = - \\delta\\Lambda^I \\beta_j (P_I)^j_{\\enskip i}\\,, \n\\end{equation}\nwhere $\\delta\\Lambda_{I}$ are constant (infinitesimal) parameters with\nthe index $I$ running over the rank of $G$ and $P_{I}$\nis a representation of the Lie algebra ${\\cal G}$ of $ G $: \n$$ \\Big[P_{[I},P_{J]}\\Big] = f_{[I J]}\n^{K}P_{K}$$ where $f_{[I J]}^{K}$ are the structure\n constants of $ {\\cal G}$ and suppose that $\\Phi(\\gamma)$ is scalar, that is \n\\begin{equation}\n\\Phi_{i}(P_{I})^{i}_{\\enskip j}\\gamma^{j} = 0\n\\end{equation}\nThen the classical $G$-current \n\\begin{eqnarray}\nJ^{0}_{I} =\\beta_{i}(P_{I})^{i}_{\\enskip j} \\gamma^{j},\n\\label{14}\n\\end{eqnarray}\nis conserved and gauge invariant.\nAlso, one can have a \"ghost\" current if it is possible \nto assign a ghost number $ g^{(i)}$ \nto each $\\gamma^{i}$ in such a way\nthat $\\Phi$ has ghost number $ 2 g^{(0)} $\n\\begin{eqnarray}\n\\sum g^{(i)}\\gamma^{i} \\Phi_{i} = 2 g^{(0)}\\Phi(\\gamma).\n\\label{15}\n\\end{eqnarray}\nThen the classical current \n\\begin{eqnarray}\nJ^0 = \\sum g^{(i)} \\beta_i \\gamma^i,\n\\label{16}\n\\end{eqnarray}\nbecomes gauge invariant and conserved. Furthermore, if $\\Phi(\\gamma)$ is homogeneous in $\\gamma$, the ghost current reads\n\\begin{eqnarray}\nJ^0 = \\beta_i \\gamma^i.\n\\label{17}\n\\end{eqnarray}\nIn order to define the basic OPE (\\ref{basic}), we choose a constant vector\n$v^i$ and define \n\\begin{eqnarray}\nY^i = \\frac{v^i}{(v^j \\Phi_j)},\n\\label{18}\n\\end{eqnarray}\n{}from which we have a relation $Y^i \\Phi_i = 1$.\nAt this stage, the basic OPE is given by\n\\begin{equation} \\label{basic2}\n<\\beta_{i}(y) \\gamma^{j}(z)> = {{1}\\over{y-z}}( \\delta_{i}^{\\enskip j} -\n\\Phi_{i}(z)Y^{j}(z)) \\equiv {{1}\\over{y-z}}( \\delta_{i}^{\\enskip j} - \nK_{i}^{\\enskip j}(z)). \n\\end{equation}\nOne can then check immediately that \n\\begin{eqnarray}\n<\\beta_{i}(y)\\Phi(\\gamma(z))> = 0.\n\\label{19}\n\\end{eqnarray}\nNow the problem is to understand in which cases this formalism is consistent,\nin other words, when it is possible to add $Y$-dependent terms to the relevant\n(gauge-invariant) operators in such a way that the corresponding OPE's are\nfree of $Y$-dependent contributions. \n\n\\subsection{$\\beta-\\gamma$ models with quadratic constraint}\n\nBefore discussing the general case, let us consider a simpler case: a class of \nmodels with the quadratic constraint \n\\begin{equation}\n\\Phi(\\gamma) = \\frac{1}{2} \\gamma^{i}g_{ij}\\gamma^{j}= 0, \n\\end{equation}\n where $g_{ij}$ is a constant, invertible, $N \\times N$ matrix with inverse\n $g^{ij}$. The matrices $g_{ij}$ and $g^{ij}$ can be used for raising and \nlowering indices, for instance, $\\gamma_{i} = g_{ij}\\gamma^{j}$.\nThen the basic OPE is\n\\begin{equation}\\label{basic3}\n< \\beta_{i}(y)\\gamma^{j}(z)> = {{1}\\over{y-z}} (\\delta_{i}^{\\enskip j} - \nK_{i}^{\\enskip j}(z)),\n\\end{equation}\nwhere \n\\begin{equation}\nK_{i}^{\\enskip j} = \\gamma_{i} Y^{j}, \n\\end{equation}\nthereby we can prove $<\\beta_{i}(y) (\\gamma^{j}g_{jk}\\gamma^{k})(z)> = 0$.\n\nThese models have the local symmetry \n\\begin{equation}\n\\delta \\beta_i = \\lambda \\gamma_i, \n\\end{equation}\nand the classically gauge-invariant composite fields are the currents\n$J^0= \\beta_{i}\\gamma^{i}$, $ J^0_{rs} = \\beta_{[r}\\gamma_{s]}$ and\nthe stress-energy tensor $ T^0 = \\beta_{i}\\partial \\gamma^{i}$.\\par\n\nWe shall utilize the $Y$-formalism to compute the OPE's for the corresponding \nquantum operators $J$, $J_{rs}$ and $T$, which are corrected by $Y$-dependent terms in order to have $Y$-independent OPE's.\nBy computing the OPE between $J^0_{rs}$ and $J^0_{pq}$ one obtains\n\\begin{eqnarray}\n< J^0_{rs}(y) J^0_{pq}(z)> &=& - {{1}\\over{(y-z)^{2}}}[ g_{sp}g_{rq} \n- K_{ps}(y)g_{qr} - K_{rq}(z)g_{sp} + K_{ps}(y)K_{rq}(z)]\n\\nonumber\\\\\n&-& {{1}\\over{y-z}} [\\beta_{r} g_{ps}\\gamma_{q} - \\beta_{p}g_{rq}\\gamma_{s}], \n\\label{ope1}\n\\end{eqnarray}\n(here antisymmetrization among $r, s$ and $p, q$ is implicitly understood)\nwhich contains spurious $Y$-dependent poles. However, it can be verified by \ncomputing the OPE's of $J^0_{rs}$ with $Y_{[r}\\partial \\gamma_{s]}$ and\n$ \\partial Y_{[r}\\gamma_{s]}$ that if one defines\n\\begin{eqnarray}\\label{current}\nJ_{rs} = J^{0}_{rs} - Y_{[r}\\partial \\gamma_{s]} - {{1}\\over{2}} \\partial \nY_{[r} \\gamma_{s]},\n\\end{eqnarray}\nthe spurious terms are precisely canceled and one gets the $Y$-independent OPE\n\\begin{eqnarray}\\label{opecurrent}\n = {{1}\\over{(y-z)^{2}}}g_{r[p}g_{q]s} + {{1}\\over{y-z}}\n(g_{p[r}J_{s]q} - g_{r[p}J_{q]s}).\n\\end{eqnarray}\nIn a similar way, provided that one defines \n\\begin{eqnarray}\\label{currentbis}\nJ = J^{0} - {{3}\\over{2}} \\partial Y_{i}\\gamma^{i},\n\\end{eqnarray}\nand \n\\begin{eqnarray} \\label{currentter}\nT = T^{0} + {{1}\\over{2}} \\partial (Y_{i}\\partial \\gamma^{i}),\n\\end{eqnarray}\nthe spurious $Y$-dependent poles also cancel in the OPE's of $J$ and $T$ with $J_{rs}$. Moreover, it turns out that the remaining OPE's of $J$ and $T$ among them are also free of spurious, $Y$-dependent contributions. \n\nTo summarize, the algebra among currents in addition to (\\ref{opecurrent})\n takes the form\n\\begin{eqnarray}\n = 0,\n\\label{qcurrent1}\n\\end{eqnarray}\n\\begin{eqnarray}\n = {{1}\\over{(y-z)^2}}J_{rs}(y),\n\\label{qcurrent2}\n\\end{eqnarray} \n\\begin{eqnarray}\n = {{N-1}\\over{(y-z)^{4}}} + {{1}\\over{(y-z)}^{2}}(T(y) + T(z)),\n\\label{qcurrent3}\n\\end{eqnarray}\n\\begin{eqnarray}\n = {{N-2}\\over{(y-z)^{3}}} + {{1}\\over{(y-z)^{2}}}J(y),\n\\label{qcurrent4}\n\\end{eqnarray}\n\\begin{eqnarray}\n = {{4 - N}\\over{(y-z)^{2}}}.\n\\label{qcurrent5}\n\\end{eqnarray}\nOne should notice that in the OPE's $$ and $$, there are $Y$-independent contributions, given by $ {3 \\over (y-z)^2} $ and ${{-1} \\over {(y-z)^{3}}}$ respectively, coming from the $Y$-dependent terms in the definition of $J$ and $T$.\n\n\\subsection{An obstruction for models with constraints of degree greater \nthan two}\n\nNow let us consider the general case where the constraint (\\ref{constraint2})\nis a polynomial in $\\gamma^i$ of order greater than two. We shall show that in \nthis case the $Y$-formalism does not work in general. To see why, let us compute\nthe OPE $$.\nThe most general expression for $J_{rs}$ is\n\\begin{eqnarray} \\label{current2}\nJ_{rs} = J^{0}_{rs} -c_{1} Y_{[r}\\partial \\Phi_{s]} - c_{2} \\partial \nY_{[r} \\Phi_{s]},\n\\end{eqnarray}\nwhere $c_{1}$ and $c_{2}$ are constants. Here $\\Phi_{ij}$ are used for lowering indices such as $Y_i = Y^j \\Phi_{ji}$. But the problem arises because the OPE's of \n$J^0_{rs}$ with the $Y$-dependent terms $Y_{[i}\\partial \\Phi_{j]}$ and \n$\\partial Y_{[i} \\Phi_{j]}$ yield poles with residuum proportional to \n$Y^{k} \\Phi_{k i p}$ which cannot \nbe canceled. \nIndeed, we can easily derive \n\\begin{eqnarray}\n = -{{1}\\over{y-z}}\nY_{[r} K_{s]i} + {1 \\over{y-z}} Y^{j} \\Phi_{[s} \\Phi_{r]ji}.\n\\end{eqnarray}\nAs before, for instance, if one selects $c_{1} = 1$ and \n$c_{2} = {{1} \\over {2}}$ in order to cancel the $Y$-dependent double \npoles, one has the equation\n\\begin{eqnarray}\n&{}& \n= {{-1} \\over{(y-z)^2}} [ \\Phi_{ps}(y) \\Phi_{qr}(z) + ( \\Phi_s \\Phi_q Y^i\n\\Phi_{ipr} )(z) ]\n+ {1 \\over{y-z}}[ \\Phi_{qr} J_{ps} - \\Phi_{ps}J_{rq} ]\n\\nonumber\\\\\n&+& {1 \\over{y-z}} [ - Y^i \\Phi_{ipr} \\Phi_s \\partial \\Phi_q + Y^{i}Y^{j}\n\\partial \\gamma^{k}(\\Phi_{ir}\\Phi_{jkp} - \\Phi_{ip}\\Phi_{jkr})\n- \\frac{1}{2} \\partial (Y^i \\Phi_{ipr}) \\Phi_s \\Phi_q], \n\\label{ope3}\n\\end{eqnarray}\nwhich is obviously inconsistent unless \n\\begin{eqnarray}\nY^{i} \\Phi_{ijk} = 0.\n\\end{eqnarray}\n(In (\\ref{ope3}), antisymmetrization among $r, s$ and $p, q$ is understood.)\n\nOne could think that the problem might depend on our choice of $Y^{i}$ and a \ndifferent choice would avoid the problem, but it is not so.\nThe most general possibility is to start with a constant vector $v_{i}$ and \nto define $Y^{i}$ as $ Y^{i}= {{v_{j} A^{j i}(\\gamma)} \\over {(v_{j} \nA^{j i}(\\gamma)\\Phi_{i})}}$. If $\\Phi_{i j}$ is invertible, calling \n${\\tilde \\Phi}^{i j}$ its inverse, one can choose $ A^{ij} = {\\tilde \\Phi}^{ij}$ so that \n$Y_{i} = {{v_{i}} \\over {(v_{j}{\\tilde \\Phi}^{ji}\\Phi_{i})}} $. However, \nthe problem remains in general since in this case \n\\begin{eqnarray}\n = \n-{{1}\\over{y-z}} Y_{[r} K _{s] i} \n+ {1 \\over{y-z}} Y_i Y^j \\Phi_{[s} \\Phi_{r]j k} {\\tilde \\Phi}^{k l}\\Phi_l.\n\\label{A1}\n\\end{eqnarray}\n\nA possible exception is the case where the constraint is homogeneous of degree\n$h$, for which we have following identities:\n\\begin{eqnarray}\n\\Phi_{i}\\gamma^i &=& h \\Phi, \\nonumber\\\\\n\\Phi_{i j}\\gamma^j &=& (h - 1) \\Phi_{i}, \\nonumber\\\\\n\\Phi_{i j k}\\gamma^{k} &=& (h - 2) \\Phi_{i j}.\n\\end{eqnarray}\nThen, we have relations\n\\begin{eqnarray}\n\\gamma^j &=& (h-1) \\tilde\\Phi^{ji}\\Phi_{i} , \\nonumber\\\\\nY_{i} &=& (h-1) {{v_{i}}\\over {(v_{j}\\gamma^{j})}}.\n\\label{A2}\n\\end{eqnarray}\nThus, using Eq's. (\\ref{A1})-(\\ref{A2}), one finds\n\\begin{eqnarray}\n = -{{1}\\over{y-z}}{{1}\\over{h-1}}\nY_{[r} K_{s]i}.\n\\end{eqnarray}\nNote that the OPE $$ is given by \n\\begin{eqnarray}\n &=& - \\frac{1}{(y-z)^2} [ \\Phi_{rq}(z)\n\\Phi_{ps}(y) - \\Phi_{rq}(z) K_{ps}(y) - K_{rq}(z) \\Phi_{ps}(y)\n+ K_{rq}(z) K_{ps}(y) ]\n\\nonumber\\\\\n&-& \\frac{1}{y-z}(\\beta_r \\Phi_q \\Phi_{ps} - \\beta_p \\Phi_s \\Phi_{rq}).\n\\end{eqnarray}\nOn the other hand, one can calculate\n\\begin{eqnarray}\n&{}& \n+ < (\\partial\\Phi_{[r}Y_{s]})(y) J^0_{pq}(z) > \n= {{-1}\\over{(y-z)^{2}}}[ \\Phi_{rq}(z) K_{ps}(y) + \\Phi_{ps}(y) K_{rq}(z)] \n\\nonumber\\\\\n&+& {{1}\\over{y-z}} [\\Phi_{rq} \\partial \\Phi_p Y_s - \n\\Phi_{ps} \\partial \\Phi_r Y_q] \n+ {{1}\\over{y-z}} [K_{rq} \\partial \\Phi_p Y_s\n- K_{ps} \\partial \\Phi_r Y_q ] {{1}\\over{h-1}},\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n&{}& < J^0_{rs}(y)(\\Phi_{[p}\\partial Y_{q]})(z)> + < (\\Phi_{[r} \\partial\nY_{s]})(y) J^0_{pq}(z) > \n= {{2}\\over{(y-z)^{2}}} K_{rq}(z) K_{ps}(y) {{1}\\over{h-1}}\n\\nonumber\\\\\n&-& {{2}\\over{y-z}} [ K_{rq} \\partial \\Phi_p Y_s \n- K_{ps} \\partial \\Phi_r Y_q ] {{1}\\over{h-1}}\n+ {{1}\\over{y-z}} [ \\Phi_{qr} \\Phi_p \\partial Y_s - \n\\Phi_{ps} \\Phi_r \\partial Y_q ].\n\\end{eqnarray}\n(In the above, antisymmetrization among $r, s$ and $p, q$ is implicitly understood again.)\nTaking $c_{1} = 1$ and $c_{2} = {{1} \\over {2}} (h-1)$ in (\\ref{current2}) \nin order to cancel the $Y$-dependent terms in the double pole, one gets\n\\begin{eqnarray}\n &=& {{-1}\\over{(y-z)^{2}}} \\Phi_{ps}(y) \\Phi_{qr}(z) +\n{{1}\\over{y-z}}( \\Phi_{qr}J_{ps} - \\Phi_{ps}J_{rq})\n\\nonumber\\\\\n&+& \\frac{1}{y-z} \\frac{2-h}{h-1} [ K_{rq} \\partial \\Phi_p Y_s \n- K_{ps} \\partial \\Phi_r Y_q ].\n\\end{eqnarray}\nThis result is consistent only if $h=2$, namely, if the constraint is \nquadratic. \n\n\\subsection{Models with partly quadratic constraint}\n\nIt can happen that for special models and with suitable choice\nof $Y^i$, the problem pointed out in the previous subsection \nis absent even if the constraint $\\Phi$ is more than quadratic in $\\gamma^i$.\n\nIndeed, suppose that the fields $\\gamma^{i}$ can be splitted in two sets\n\\begin{eqnarray}\n\\gamma^i = ( \\gamma^{a} \\equiv y^{a}, \\gamma^{\\hat a}\\equiv x^{\\hat a}),\n\\end{eqnarray}\nwhere $a = 1, \\cdots, n_1$, $\\hat a = 1, \\cdots, n_2$ ($n_{1} + n_{2} = N$), and that the constraint can be written as \n\\begin{equation} \\label {newconstraint}\n\\Phi(y,x) \\equiv \\Phi^{(1)}(y) + \\Phi^{(2)}(x) = 0, \n\\end{equation}\nwhere \n\\begin{eqnarray}\n\\Phi^{(1)}(y) = \\frac{1}{2} y^a g_{ab} y^b,\n\\end{eqnarray}\nis quadratic in $y^{a}$ while $\\Phi^{(2)}(x)$ is a generic polynomial in $x^{\\hat a}$.\nThe $Y$-formalism can be adapted to this class of models by choosing \n$Y^{i}$ (that is, $v^{i}$) to have non-vanishing components only in \nthe direction of $y^a$\n\\begin{eqnarray}\nY^i = Y^a \\delta_a^i.\n\\end{eqnarray}\nWith this choice, the formalism turns out to be consistent. In fact, in this case, the condition $Y^{i} \\Phi_{ijk} = 0$ \nis satisfied. \nThe action is\n\\begin{eqnarray}\nS = \\int \\beta_{a}\\bar\\partial y^{a} \n+ \\int \\beta_{\\hat b}\\bar\\partial x^{\\hat b},\n\\end{eqnarray}\nand the model has the local symmetry \n\\begin{eqnarray}\\label{gaugeA}\n\\delta \\beta_i = \\lambda \\Phi_i(x, y).\n\\end{eqnarray}\nThe classically gauge-invariant operators are \n\\begin{eqnarray}\nT^0 &=& \\beta_i \\partial \\gamma^i, \\nonumber\\\\\nJ^0_{ij} &=& \\beta_{[i} \\Phi_{j]}.\n\\end{eqnarray}\nIf the constraint $\\Phi(x, y)$ is generic, $J_{i j}$ and $T$ are \nthe only gauge-invariant operators, but there are further \ngauge-invariant operators for a specific form of constraints.\n\nIn order to observe this fact, let us suppose that one can assign a \n\"ghost number\" $g^{(a)}$ and $g^{(\\hat b)}$ to $y^{a}$ and $x^{\\hat b}$, respectively, in such a way that both $\\Phi^{(1)}\n(y)$ and $\\Phi^{(2)}(x)$ have ghost number $2g^{(0)}$.\nThen, the classical \"ghost current\"\n\\begin{eqnarray}\nJ^0 = \\sum g^{(a)}\\beta_{a}y^{a} + \\sum g^{(\\hat b)}\\beta_{\\hat b} \nx^{\\hat b},\n\\end{eqnarray}\nis gauge invariant and conserved.\n\nMoreover, if $y^a$ and $x^{\\hat b}$ belong to two representations\nof a Lie group $G$\n\\begin{eqnarray}\n\\delta y^{a} = \\delta \\Lambda^{I} (P_{(1)I})^{a}_{\\enskip b}y^{b},\n\\qquad \\delta x^{\\hat a} = \\delta\\Lambda^{I} \n(P_{(2)I})^{\\hat a}_{\\enskip\\hat b}x^{\\hat b},\n\\end{eqnarray}\nand $\\Phi^{(1)}(y) $ and $ \\Phi^{(2)}(x)$ are scalars, the classical\n$G$-current \n\\begin{eqnarray}\nJ^0_{I} = \\beta_{b}( P_{(1)I})^{b}_{\\enskip \na} y^{a} + \\beta_{\\hat b}( P_{(2) I})^ {\\hat b}_{\\enskip \\hat a} x^{\\hat a},\n\\end{eqnarray}\nis also gauge invariant and conserved. \nNotice that with the constraint (\\ref{newconstraint}) the currents \n$J^0_{(1) I} = \\beta_{b}( P_{(1)I})^{b}_{\\enskip a} y^{a}$ and $J^0_{(2) I} \n= \\beta_{\\hat b}( P_{(2) I})^{\\hat b}_{\\enskip \\hat a} x^{\\hat a} $ are separately gauge invariant and conserved.\n\nIt is always possible to redefine $g^{(a)}$ as $ g^{(a)} = g^{(0)} + q^{(a)}$\n so that $\\sum q^{(a)}y^{a}g_{ab}y^{b} = 0 $ ; then the ghost current becomes\n\\begin{eqnarray} \\label{ghostcurrent2}\nJ^0 = g^{(0)} \\sum \\beta_{a}y^{a} + \\sum g^{(\\hat b)}\\beta_{\\hat b} \nx^{\\hat b}.\n\\end{eqnarray}\nMoreover, if it is possible to assign charges $q^{(\\hat a)}$ to $x^{\\hat a}$\nsuch that $\\sum q^{(\\hat a)} x^{\\hat a}\\Phi^{(2)}_{\\hat a} = 0 $, \n$\\hat J^0 = \\sum q^{(a)}\\beta_{a}y^{a} + \\sum q^{(\\hat b)}\\beta_{\\hat b}\n x^{\\hat b}$ is gauge invariant and conserved.\n$\\hat J^0$ is a particular $G$-current with $G= U(1)$. Notice that if \n$\\Phi^{(2)}$ is homogeneous of degree $h > 0$, one can write $ g^{(\\hat b)} = {{2}\\over{h}}\ng^{(0)} + q^{(\\hat b)}$ and the ghost current (rescaled to have $g^{(0)}= 1$) \nbecomes\n\\begin{eqnarray}\nJ^0 = \\sum \\beta_{a}y^{a} +{{2}\\over{h}} \\sum \\beta_{\\hat b} x^{\\hat b}.\n\\end{eqnarray}\n\nNow we want to determine the $Y$-dependent correction terms in these currents\nsuch that the spurious $Y$-dependent terms do not appear in the OPE's among\nthese currents. In addition, we wish to compute these OPE's.\n\nIn this case, the basic OPE \n\\begin{eqnarray}\n<\\beta_{i}(y)\\gamma^{j}(z)> = {{1}\\over{y-z}}(\\delta_{i}^{\n\\enskip j} - \\Phi_{i}(z)Y^{j}(z)),\n\\end{eqnarray}\nsplits as follows:\n\\begin{eqnarray}\n<\\beta_{a}(y) y^{b}(z)> &=& {{1}\\over{y-z}}(\\delta_{a}^{\\enskip b} - \ny_{a}(z)Y^{b}(z)),\n\\nonumber\\\\\n<\\beta_{a}(y) x^{\\hat b}(z)> &=& 0,\n\\nonumber\\\\\n<\\beta_{\\hat a}(y) x^{\\hat b}(z)> &=& {{1}\\over{y-z}}\n\\delta_{\\hat a}^{\\enskip \\hat b},\n\\nonumber\\\\\n<\\beta_{\\hat a}(y) y^{b}(z)> &=& - {{1}\\over{y-z}}\n\\Phi_{\\hat a}(z) Y^{b}(z),\n\\end{eqnarray}\nwhere we have defined as $y_{a} = g_{ab} y^{b}$.\nAs a preliminary, let us coinsider the OPE's \n\\begin{eqnarray}\n<\\beta_j(y) Y^k(z)> &=& - {{1}\\over{y-z}} [ Y_j Y^k \n- \\Phi_j (Y^l Y_l) Y^k ],\n\\nonumber\\\\\n<\\beta_j(y) Y_k(z)> &=& - {{1}\\over{y-z}} [ Y_j Y_k \n- \\Phi_j (Y^l Y_l) Y_k].\n\\end{eqnarray}\nThe terms proportional to $\\Phi_j (Y^l Y_l)$ do not contribute \nto the OPE's of $Y$ and $\\partial Y$ with the currents $J_{ij}$, $T$ and the \nother currents when they exist. In fact, in the OPE of $\\beta_{[i}\n\\Phi_{j]}$ with $Y$ and $\\partial Y$, they give rise to terms \nproportional to $\\Phi_{[i} \\Phi_{j]}$ and $ \\partial(\\Phi_{[i}\n\\Phi_{j]})$.\nIn the OPE of $\\sum g^{(i)} \\beta_i \\gamma^i$\nwith $Y$ and $\\partial Y$, they give rise to terms proportional to \n$\\sum g^{(i)} \\Phi_i \\gamma^i = 2 g^{(0)} \\Phi = 0$ and to \n$\\partial\\Phi = 0$. In the OPE of $\\beta_{i} \\partial\\gamma^{i}$, we\nhave terms proportional to $\\Phi_i \\partial \\gamma^i = \\partial \\Phi = 0$ \nand to $\\partial^2 \\Phi = 0$ and so on.\n\nMoreover, let us notice that \n\\begin{eqnarray}\n< \\sum (g^{(i)} \\beta_i \\gamma^i)(y) \\Phi_j(z)> \n&=& {{1}\\over{y-z}} \\sum g^{(i)} \\gamma^i \\Phi_{ji}\n\\nonumber\\\\\n&=& {{1}\\over{y-z}} [\\partial_{j} (\\sum g^{(i)} \\gamma^i \\Phi_i) \n- \\sum g^{(i)} \\delta_j^i \\Phi_i] \n\\nonumber\\\\\n&=& {{1}\\over{y-z}}(2 g^{(0)} - g^{(j)}) \\Phi_j.\n\\end{eqnarray}\nGiven that, the remaining calculations are straightforward. It then turns out that, in agreement with (\\ref{current}) and (\\ref{currentter}), the $Y$-dependent corrections for $J_{i j}$ and $T$ are given by \n\\begin{eqnarray}\nJ_{ij} &=& J^0_{ij} - Y_{[i}\\partial \\gamma_{j]} - {{1}\\over{2}} \\partial \nY_{[i} \\gamma_{j]},\n\\nonumber\\\\\nT &=& T^0 + \\frac{1}{2} \\sum_a \\partial (Y_a \\partial y^a).\n\\end{eqnarray}\nThe OPE's among these operators are\n\\begin{equation}\n \n= {{1} \\over{2(y-z)^2}} [ \\Phi_{k[i}(y) \\Phi_{j]l}(z)\n- \\Phi_{l[i}(y) \\Phi_{j]k}(z) ]\n+ {1 \\over{y-z}}[ \\Phi_{k[i} J_{j]l} - \\Phi_{l[ i}J_{j]k} ],\n\\end{equation}\n\\begin{eqnarray}\n = {{1}\\over{(y-z)^{2}}} J_{ij}(y),\n\\end{eqnarray}\n\\begin{eqnarray}\n = {{N - 1}\\over{(y-z)^4}} + {{1}\\over{(y-z)^2}}(T(y) + T(z)).\n\\end{eqnarray}\n\nAs for the current $J$, when it exists, one gets\n\\begin{equation}\n J = J^0 +3\/2 g^{(0)} \\sum Y_{a}\\partial y^{a},\n\\end{equation}\nwhere $J^0$ is given in (\\ref{ghostcurrent2}).\nThen the relevant OPE's read\n\\begin{eqnarray}\n = {{1\/2}\\over{(y-z)^{2}}} (g^{(j)} - g^{(i)})\\Phi_{i j}\n+ (2g^{(0)}-g^{(i)} - g^{(j)}){{1}\\over{y-z}} J_{i j},\n\\end{eqnarray}\n\\begin{eqnarray}\n = {{1}\\over{(y-z)^3}} (\\sum g^{(i)} - 2 g^{(0)}) +\n {{1}\\over{(y-z)^2}}J(y),\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n = {{1}\\over{(y-z)^{2}}} (4 (g^{(0)})^{2} - \\sum g^{( i)}g^{(i)}).\n\\end{eqnarray}\nIf the constraint allows for a $G$-current $J^0_{I} = \\beta_{i}(P_{I})^{i}_{\n\\enskip j}\\gamma^{j}$ one finds that the corrected form of this current is\n\\begin{eqnarray}\nJ_{I} = (\\beta P_{I} \\gamma) - (Y P_{I} \\partial \\gamma) - 1\/2 (\\partial Y P_{I}\\gamma),\n\\end{eqnarray}\nand the relevant OPE's are\n\\begin{equation}\n< J_{I}(y) J_{K}(z)> = - {{1}\\over{(y-z)^2}} Tr(P_I P_K) + {{1}\\over{y-z}} f_{KI}^{\\enskip J}J_{J}, \n\\label{J-algebra2}\n\\end{equation}\n\\begin{equation}\n< J_{I}(y) J_{i j}(z)> = {{1}\\over{2 (y-z)^2}} \n[\\Phi_{il}(P_{I})^l_{\\enskip j} \n- \\Phi_{jl}(P_{I})^l_{\\enskip i}]\n+ {{1}\\over{y-z}}[ (P_{I})_{i}^{\\enskip l}J_{l j} -\n (P_{I})_{j}^{\\enskip l}J_{l i} ],\n\\end{equation}\n\\begin{equation}\n = {{1}\\over{(y-z)^3}}Tr(P_{I}) + {{1}\\over{(y-z)^2}} \nJ_{I}(y),\n\\end{equation}\nwhen $ Tr(P_{I})$ is different from zero only if $G$ contains $U(1)$-factors.\n\n\n\\section{Examples}\n\\subsection{Projective Spaces}\n\nAn interesting class of $\\beta-\\gamma$ systems, either constrained or \nunconstrained, are the gauged $\\beta-\\gamma$ models. They appear for two \nreasons: the first one is the construction of model defined on a space whose \ndescription is given in terms of some free coordinates modulo some gauge symmetry. For \nexample the projective space which can be formulated as quotients of flat $n$ dimensional \ncomplex spaces modulo some toric gauge symmetries. The second reason is that in some \ncase hypersurfaces can be viewed as manifolds with some gauge symmetries. \nObviously, this relation has been used in literature \nfor many important results (see for example \\cite{Hori:2003ic}), but it has not been developed for \n$\\beta-\\gamma$ systems (some considerations can be found in \\cite{Tan:2006zg}). The $Y$-formalism can be also used \nin the present context, where first one uses the gauge symmetry to remove unwanted coordinates and \nthen uses the $Y$-formalism to compute the remaining OPE's. \n\nLet us consider a group $G$ of rigid transformations: \n\\begin{equation}\n\\delta \\gamma^i = \\delta\\Lambda^I (P_I)^i_{\\enskip j}\\gamma^{j}\\,, ~~~~\n\\delta \\beta_i = - \\delta\\Lambda^I \\beta_j (P_I)^j_{\\enskip i}\\,, \n\\end{equation}\nunder which, not only the action trivially, but also the constraint, if it \nexists, is invariant.\n\nWe can promote the constant parameters $\\delta\\Lambda_I$ to local ones, by \nadding to the action a set of gauge fields $\\bar A^I$ coupled to the currents \n(constraints)\n$J_{I} = \\beta_{i} (P_{I})^{i}_{\\enskip j}\\gamma^{j}$.\nEventually one can also add to the action a topological $B-F$ term\n$\\int \\pi^{I}(\\bar \\partial A - \\partial \\bar A - f^{I}_{\\enskip JK}A^{J}\\bar \nA^{K})$. \n\nFor our purposes we are mainly interested in the abelian case.\nIn particular, if the constraint, when it exists, is homogeneous, one can gauge\nthe scale symmetry \n\\begin{equation} \\label{scale}\n\\gamma^i \\rightarrow \\Lambda \\gamma^i\\,, ~~~~\n\\beta_i \\rightarrow \\Lambda^{-1} \\beta_i,\n\\end{equation}\ngenerated by the current $ J= \\beta_{i}\\gamma^{i}$ and\n under which the gauge field transforms as $\\bar A \\rightarrow \\bar A + \n\\bar\\partial \\Lambda $.\nThese models describe projective spaces. \nFor instance, if $\\gamma^i$ \nparametrize the complex plane ${\\bf C}^{n+1}$, the gauge symmetry projects \non the projective space ${\\bf CP}^n$. \n\nIn the presence of a (homogenous) constraint the model describes a \nhypersurface in a projective space. A discussion on the subject is \ncontained in the paper \\cite{Grassi:2007va}. \n \n\n\nAs a first example, we consider the torus. \nIt can be viewed as a cubic hypersurface in ${\\bf P}^2$ of the form\n\\begin{equation} \\label{full}\n(\\gamma^1)^3 = \\gamma^2 (\\gamma^2 -\\gamma^3)(\\gamma^2 - a \\gamma^3),\n\\end{equation}\nwhere $a$ is the modulus of the torus. The hypersurface is homogeneous of \ndegree 3\nwith respect to the rescaling of the fields $\\gamma^i \\sim \\Lambda \\gamma^i$.\n The gauge symmetry can be used \nto gauge to one the coordinate $\\gamma^2$ and the constraint becomes \n\\begin{equation}\\label{reduced}\n(\\gamma^1)^3 = (\\gamma^3 -1)(a \\gamma^3 - 1)\\,,\n\\end{equation}\nand the model can be treated as constrained in a reduced space. \n\nTo be more explicit, following the general recipe for gauge-fixing procedure,\n we must add a single ghost field $c$ with the BRST transformations \n\\begin{equation}\ns\\, \\gamma^i = c \\gamma^i\\,, ~~~\ns\\, \\beta_i = - c \\beta_i\\,, ~~~\ns\\, \\bar A = - \\bar \\partial c\\,, ~~~\ns\\, c =0\\,, ~~~\ns\\, b = \\rho\\,, ~~~\ns\\, \\rho =0 \\,.\n\\end{equation}\nThen, among many of possible gauge-fixing conditions, two of them are more \ninteresting. \nThe first one is to add to the action $S_A = \\int (\\beta_{i}\\bar\\partial\n\\gamma^{i} + \n\\bar A \\beta_{i} \\gamma^{i}) $ the BRST exact term $ \\int s(b \\bar A)$\nto reach the gauge $\\bar A = 0 $. The result is the action\n\\begin{equation} \nS = \\int (\\beta_{i}\\bar\\partial\\gamma^{i} + b \\bar \\partial c),\n\\end{equation}\nwith the constraint (\\ref{full}) and a unbroken, rigid scale invariance.\n\nIn the second gauge-fixing, one adds to the action the BRST variation of the \ngauge fermion $ b(\\gamma^2 - 1) $, that is, $\\int s( b (\\gamma^2 - 1))\n $ that leads to the gauge $\\gamma^2 = 1 $ so that the action becomes\n\\begin{eqnarray}\\label{gauge-fixedaction}\nS = \\int \\Big(\\beta_{i}\\bar\\partial\\gamma^{i} + \\bar A \\beta_{i} \\gamma^i\n + \\hat \\rho (\\gamma^2 - 1) - b c\\Big), \n\\end{eqnarray}\nwhere $\\hat \\rho = \\rho - bc$.\nThen, integrating over $ \\hat \\rho$, one gets the gauge-fixing condition, \nintegrating over the gauge field $\\bar A$, one gets that \n$\\beta_2 = - \\sum_{i\\neq 2} \\beta_i \\gamma^i$ and integrating over $b$, \none obtains $c = 0 $. The action is still free and the result is\n\\begin{eqnarray}\\label{gauge-fixedaction2}\nS = \\int \\Big(\\sum_{i\\neq 2}\\beta_{i}\\bar\\partial\\gamma^{i} \\Big) \\,,\n\\end{eqnarray}\nwith the constraint (\\ref{reduced}). In this case as well, the rigid scale invariance is unbroken.\n \n\n\\subsection{The conifold}\n\nThe second example is the conifold constraint and the \nrelated Lie algebra of $D_2 = A_1 \\times A_1$. This example has been already discussed \nin the introduction and it is based on the constraint (\\ref{conio}). According to the $Y$-formalism, \n the central charge is $3$ and the level of the currents $J_{ij} $ is $-1\/2$.\n Moreover the OPE's of the current $J$ \nare given by\n\\begin{equation}\n = 0\n\\end{equation} \n\\begin{eqnarray}\n = {{2}\\over{(y-z)^{3}}} + {{1}\\over{(y-z)^{2}}}J(y),\n\\label{qcurrent4coni}\n\\end{eqnarray}\n\\begin{eqnarray}\n = 0.\n\\label{qcurrent5coni}\n\\end{eqnarray}\nTo check these results we study the model in two ways.\n \nIn the first way, we use a description similar to that given in \\cite{Berkovits:2005hy} and \nin the second way, we use the description as a gauged linear sigma model. \n\nOn a patch where $\\gamma^1\\neq 0$, we solve the conifold constraint with respect to \nthe coordinate $\\gamma^2 = \\gamma^3\\, \\gamma^4\/ \\gamma^1$. Then, we redefine the coordinates as follows:\n\\begin{equation}\\label{coniA}\n\\gamma^1 = \\gamma\\,,~~~~~~~\n\\gamma^3 = \\gamma u\\,,~~~~~~~\n\\gamma^4 = \\gamma v\\,,~~~~~~~\n\\end{equation}\nand this yields $\\gamma^2 = \\gamma uv$. Then, inserting these definitions in the action we can \nidentify the conjugate momenta to the new fields $\\gamma, u$ and $v$ by\n\\begin{equation}\n\\beta = \\beta_1 + \\beta_2 uv + \\beta_3 u + \\beta_4 v\\,, ~~~~\n\\beta_u = \\beta_2 \\gamma v + \\beta_3 \\gamma\\,, ~~~~~\n\\beta_v = \\beta_2 \\gamma u + \\beta_4 \\gamma\\,,\n\\end{equation}\nwhere $\\beta, \\beta_u$ and $\\beta_v$ are the conjugate momenta of $\\gamma, u$ and $v$, \nrespectively. The new action, obtained by these field redefinitions, is again free and therefore we \ncan use free OPE's for the computations. We have to notice that the redefinition of the fields is \nnon-linear and the total ghost charge is carried by the field $\\beta$ and $\\gamma$. \nThe next step is to translate the algebra of (gauge-invariant) currents $J_{ij}$ and $J$ into \nthe new variables. We observe that the combinations $\\beta, \\beta_u$ and $\\beta_v$ are gauge-invariant combinations under the gauge transformations discussed in (\\ref{gaugeA}) and therefore \nthe currents should be expressible in terms of those gauge-invariant basic fields. After a \nbit of algebra we get \n\\begin{eqnarray}\\label{coniB}\nJ_{12} &=& :\\beta \\gamma: - :\\beta_u u: - :\\beta_v v: \\,, ~~~~~\nJ_{13} = :\\beta_v v^2: - :\\beta \\gamma:v \\,, ~~~~~ \\nonumber \\\\\nJ_{14} &=& :\\beta_u u^2: - :\\beta \\gamma:u \\,, ~~~~~~ \nJ_{23} = - \\beta_u \\,, ~~~~~ \\nonumber \\\\\nJ_{24} &=& - \\beta_v \\,, ~~~~~ \nJ_{34} = :\\beta_v v: - :\\beta_u u: \\,, \\nonumber \\\\\nJ&=& \\beta \\gamma\\,.\n\\end{eqnarray}\nIt is easy to check that the currents $J_{ij}$ generate the $SO(4,{\\mathbf C})$, but \nthey have double poles with the current $J$. In order to decouple the currents $J_{ij}$ from \nthe ghost currents, we add to the combinations $\\beta \\gamma$ an additional piece \n${1\\over 2} \\partial \\gamma$ which can be seen as a normal ordering term. It is easy to check that the double poles generated by this new piece are \nenough to cancel the double poles between the two sets of currents and finally we can check that \nthe OPE of $J$ with itself gives level zero which is consistent with $Y$-formalism (compare with \nEq. (\\ref{qcurrent5}). \n\nThe second way of proceeding is to use the gauged linear sigma model. We \napply this technique again to the conifold case in order to illustrate some ambiguities emerging \nfrom this approach (this discussion is similar to the analysis given in \\cite{Grassi:2007va}). \nIn order to use free coordinates plus a gauge symmetry we introduce the new fields\n\\begin{equation}\n\\label{newfields}{\n\\gamma^1 = a_1 a_2\\,, ~~~~\n\\gamma^2 = a_3 a_4\\,, ~~~~\n\\gamma^3 = a_1 a_4\\,, ~~~~\n\\gamma^4 = a_3 a_2\\,, ~~~~\n} \n\\end{equation}\nwhich automatically satisfy the constraint. \nThey transform as $a_i \\rightarrow \\Lambda a_i$ for $i=1, 3$ and \n$a_i \\rightarrow a_i \/ \\Lambda$ for $i=2, 4$. So, we can rewrite the currents as follows:\n\\begin{equation}\n\\label{newcurrents}{\nJ^+_{+} = p_1 a_3\\,, ~~~~\nJ^-_{+} = p_3 a_1\\,, ~~~~\nJ^0_{+} = \\frac{1}{2} (p_1 a_1 - p_3 a_3)\\,, ~~~~\nK_+ = \\frac{1}{2} (p_1 a_1 + p_3 a_3), \n}\n\\end{equation}\nand similarly by substituting $a_1, a_3$ into $a_2, a_4$ and $p_1, p_3$ into $p_2, p_4$ and \nchanging the subindex $+$ into $-$. Notice that \nthey form an $A_1 \\times GL(1) \\times A_1 \\times GL(1)$ algebra. \n\nExpressed in terms of the variables $\\beta_{i}$ and $\\gamma^{j}$ the currents\n$J_{\\pm}^{\\pm}$, $J_{\\pm}^{0}$ are linear combinations of $J_{ij}$ and \n\\begin{eqnarray}\nK_{+} = K_{-}= \\frac{1}{2} J,\n\\end{eqnarray}\nso that \n\\begin{eqnarray}\n\\hat J = K_{+} - K_{-},\n\\end{eqnarray}\nvanishes. Then in\n the model expressed in term of the variables $p_{i}$ and $a^{i}$, $\\hat J$\nis a constraint and the model is a gauged model with action\n\\begin{eqnarray}\nS = \\int ( p_{i} a^{i} + \\bar A \\hat J ).\n\\end{eqnarray}\n$\\hat J$ is a primary field with vanishing anomaly (i.e., vanishing triple pole\nin its OPE with $T$) and \n\\begin{equation}\n\\label{ghostcurrent1}\n{<\\hat J(z) \\hat J(w)> = 0\\,.}\n\\end{equation}\nThe fact that there is no double pole in the ghost current can be \nchecked by using the Nekrasov method \n\\cite{Nekrasov:2005wg} \nstarting from the character of the zero modes $\\chi(t) = (1- t^2)\/(1-t)^4$ \nby substituting $t \\rightarrow e^x$ and expanding the result \nas a polynomial of $x$ and $\\log(x)$. Taking into account of the contribuion \nof the ghost fields $b$, $c$ coming from the gauge fixing, the central charge \nis $2$. As for the gauge current \n\\begin{eqnarray}\nJ \\equiv K_{+} + K_{-},\n\\end{eqnarray}\none has \n\\begin{eqnarray}\n &=& \\frac{2}{(z-w)^3} + {{J(z)} \\over {(z-w)^2}},\n\\nonumber\\\\\n &=& - \\frac{1}{(z-w)^2}.\n\\end{eqnarray}\nMoreover the level of the OPE among the currents that generate \n$(A_1) \\times (A_1)$ turns out to be $-1\/2$. These features agree\nwith the central charge, anomaly and levels computed using the $Y$-formalism\n(as well as using the techniques in \\cite{Berkovits:2005hy}) except for the \nOPE $ $ which vanishes in the $Y$-formalism computation.\nHowever notice that, due to the constraint, there is an ambiguity in the definition \nof the current $J$, in the model expressed in terms of $p_{i}$ and $a^{i}$.\nIndeed we can always add a term proportional to the constraint \n\\begin{equation}\nJ = (K_+ + K_-) + \\alpha (K_+ - K_-),\n\\end{equation}\n which changes the double poles (they are contact terms) but does not \n change the rest of the algebra.\n It is easy to check that the current $J$ has zero level if we set \n$\\alpha = \\pm i$. \nIt should be possible to decompose the \ncharacter formula given in \\cite{Grassi:2007va} \nin terms of the corresponding characters. Notice that all characters have \nnegative levels and therefore we expect some singular vectors of the Verma\n modules of the corresponding field theories. \n\n\\section{Adding other variables}\n\nIn general the models described above are purely bosonic and are lacking in the\n BRST charge to construct the physical space of states. Moreover they fail \nto provide a conformal field theory with zero central charge. In order to \novercome these problems, we can add new variables. The natural setting is to \nreproduce the pure spinor construction, adding some \nbosonic variable $p, X$ and some fermionic variables $p_i, \\theta^i$ with \n$p(z) X(w) \\rightarrow (z-w)^{-1}$ and $p_i(z) \\theta^j(w) \\rightarrow \n\\delta^j_i (z-w)^{-1}$.\n We assume that \nthere is only a constraint $\\Phi(\\gamma) = 0$ . The \nindex for the fermionic variables is the same as that for the bosonic\n non-linear sigma model ones, described by $\\gamma^i$. \n\nThe setting is described in paper \\cite{Grassi:2005jz}, where it has been \nshown the structure of the model, \nand the gauge-invariant operators. In \\cite{Grassi:2005jz}, a simple \nexample of constraint $\\Phi(\\gamma) = \\gamma^1 \\gamma^2$ has been taken \ninto account, but the technique can be used for more general examples \nof the form \n$\\Phi(\\gamma) = 1\/2 \\gamma^i g_{ij} \\gamma^j$. \nIn addition to the operators $J$ and $N_{i j}\\equiv J_{ij}$, \nwhich are gauge invariant under the gauge \nsymmetry $\\beta_i = \\Lambda g_{ij} \\gamma^j$, we shall consider the operators\n \\begin{equation}\\label{defs}\n\\Pi = p + {1\\over 2} \\theta^i g_{ij} \\partial \\theta^j\\,, ~~~~\nd_i = p_i + {1\\over 2} \\partial X g_{ij} \\theta^j,\n\\end{equation}\nso that\n\\begin{eqnarray}\n<\\Pi(z) \\Pi(w)> &=& 0, \\nonumber\\\\\n &=& {{1}\\over{z-w}} \\partial\\theta_{i},\n\\nonumber\\\\\n &=& {{1}\\over{z-w}} g_{i j}\\partial X.\n\\end{eqnarray}\nThen one can establish a nilpotent BRST charge in the same way as in the pure spinor formalism\n\\begin{equation}\\label{BRST}\nQ = \\oint \\gamma^i d_i.\n\\end{equation}\nNotice that the presence of the constraint $\\Phi(\\gamma)=0$ is essential to \nhave a nilpotent BRST charge. Using this \ncharge one can construct the physical space of states. \n\nAs is known, the gauge symmetry (\\ref{gaugesymm}) implies that the physical \nquantities \nshould be gauge invariant under it. This implies that only the gauge-invariant combinations containing the field $\\beta_i$ \nare $ N_{ij} \\equiv J_{ij}$, $J$ and $T$ of zero ghost number as \ndefined in (\\ref{current}), (\\ref{currentbis}) and (\\ref{currentter}), respectively. Therefore, there do not exist $Y$-independent operators with\nnegative ghost number which permit the construction \nof an operator $B$ with ghost number $-1$ such that \n\\begin{equation} \\label{brst-b}\n[Q, B] = \\hat T.\n\\end{equation}\nHere $\\hat T $ is the total energy momentum tensor of the model given by\n\\begin{eqnarray}\n\\hat T = p \\partial X + p_i \\partial \\theta^i + T,\n\\end{eqnarray}\nwhere $T$ is the energy \nmomentum tensor of the $\\beta-\\gamma$ system given by (\\ref{currentter}). \nHowever, as has been noticed in \\cite{Berkovits:2004px} for the case of pure spinors, one \ncan construct a sequence of \noperators $\\{G^i, H^{[ij]}, \\cdots\\}$ \nthat satisfy the following equations \n\\footnote{ $:\\gamma^i \\hat T :$ has the normal ordering term $ -1\/2 \n\\partial^{2} \\gamma^{i}$ (to be primary).} \n\\begin{equation} \\label{sequence}\n[Q, G^i] = :\\gamma^i \\hat T :\\,, ~~~~\n[Q, H^{[ij]}] = \\gamma^{[i} G^{j]}\\,, ~~~~ \\cdots\n\\end{equation}\nwhere\n\\begin{equation}\\label{bfield}\nG^i = \\Pi d^i +2 N^{i j}\\partial \\theta_{j} \\,, ~~~~\nH^{[ij]} = N^{i j}\\Pi \\,,~~~~\n\\gamma^{[i} H^{jk]} =0.\n\\end{equation}\nIt is intersting to observe that also in this simple model \nthis construction can be done and looks simpler than in the pure spinor case \nsince here there is no gamma matrices structure to take into considerations. \nEquations (\\ref{brst-b}) can be checked immediatly at the classical level,\nbut we have verified that they hold also at the quantum level: indeed the \n$Y$-dependent terms that arise in (\\ref{bfield}) combine in an appropriate way\nto give (\\ref{brst-b}) if one takes in account correctly the subtleties \nof the normal reordering through the so-called rearrangement theorem (see \n\\cite{Y-formalism:2007ab} for details of the procedure). \n\nWith the aid of $G^{i}$ one can construct a $Y$-dependent $B$-ghost \n\\begin{eqnarray}\nB^{(Y)} = Y_{i}G^{i},\n\\end{eqnarray}\nwhich satisfies (\\ref{brst-b}). However, with the help of both\n$G^{i}$ and $H^{ij}$ one can construct a $Y$-independent $B$-field, by \nthe so-called minimal approach, introducing the picture-changing operators \n$Z_B = [Q, \\Theta(B J + B^{ij} N_{ij})]$ and $Y_C = C_i \\theta^i \\delta(C_j \n\\gamma^j)$, and \nconstructing the field $B_B$ such that \n\\begin{eqnarray}\n[Q, B_B] = Z_B T.\n\\end{eqnarray}\nMoreover it is also possible to imitate the non-minimal approach of \n\\cite{Berkovits} by introducing $N$ quartets of new additional fields \nand get a consistent expression for $B$ (as opposed to $B_B$) in terms of the \noperators (\\ref {bfield}).\n\nThe bosonic new fields are $\\lambda_{i}$, with momenta $w^{i}$ and the fermonic\nones are $r_{i}$ with momenta $s^{i}$ with the constraints \n\\begin{equation}\n \\lambda_{i}g^{ij}\\lambda_{j}=0 \\, ~~~~ \\lambda_{i}g^{ij}r_{j}=0.\n\\end{equation} \n The new BRST charge is \n$ Q = \\oint \\gamma^i d_i + \\oint w^{i}r_{i}$ and of course the new $\\hat T$\ncontains two new terms coming from these quartets.Also\nthe $Y$-formalism can be \nextended to this sector following \\cite{Y-formalism:2007ab}.\n\nThen the $B$-field is\n\\footnote{ $:{{\\lambda_{i}}\\over{(\\gamma\\lambda)}}G^{i}:$ contains normal \nordering terms that we will not specify.}\n\\begin{equation}\n B = s^{i}\\partial \\lambda_{i} + :{{\\lambda_{i}}\\over{(\\gamma\\lambda)}}G^{i}:\n -2{{\\lambda_{i} r_{j}}\\over {(\\gamma\\lambda)^{2}}}H^{ij},\n\\end{equation}\nand satisfies (\\ref{brst-b}).\nOne can verifies that, as in the pure spinor case, this $B$-field is \ncohomological equivalent to the $Y$-dependent $B$-field, $B^{(Y)}$. \n\n\\section{Superprojective spaces}\n\\subsection{An example: the supercone constraint}\n\nWe would like to provide an example of conformal field theory \nwhere the constraint (which emerges by requiring that the BRST charge is \nnilpotent) is\n\\begin{eqnarray}\nx y + i \\theta^1 \\theta^2 = 0,\n\\label{coneA}\n\\end{eqnarray}\nwhere $x,y$ are bosonic coordinates and $\\theta^i$ are fermionic coordinates. \nThis constraint defines a supercone.\nTo derive the constraint (\\ref{coneA}) we can start from the supergroup $Osp(2|2)$. \nThis supergroup is described in terms of 4 bosonic generators $H, \\widetilde H, E^\\pm$ and \n4 fermionic ones $Q_\\alpha, Q_{\\hat \\alpha}$ with $\\alpha, \\hat \\alpha =1,2$. \nThe algebra $osp(2|2)$ is easily described by the (anti)commutators \n\\begin{eqnarray}\\label{osp}\n&&[H, E_\\pm] = \n\\pm 2E_\\pm\\,, ~~~~~\n[E^+ , E^-] = H\\,, ~~~~~\n[H, \\widetilde H] = 0\\,, ~~~~~~ \\nonumber \\\\\n&&[\\widetilde H, E_\\pm] = 0\\,, ~~~~~~~\n[\\widetilde H, Q_\\alpha ] = \\epsilon_{\\alpha\\beta} Q_{\\beta} \\,, ~~~~~\n[\\widetilde H, Q_{\\hat\\alpha} ] = \\epsilon_{\\hat\\alpha\\hat\\beta} Q_{\\hat\\beta} \\,, ~~~~~ \n\\nonumber \\\\\n&&[H, Q_\\alpha ] = Q_\\alpha\\,,~~~~~\n[H, Q_{\\hat\\alpha} ] = - Q_{\\hat\\alpha}\\,,~~~~~ \\\\\n&&\n\\{Q_\\alpha, Q_\\beta \\} = {1\\over2} \\delta_{\\alpha \\beta} E^+ \\,,~~~~~ \n\\{Q_{\\hat \\alpha}, Q_{\\hat\\beta} \\} = {1\\over2} \\delta_{\\hat\\alpha \\hat\\beta} E^- \\,,~~~~~ \n\\{Q_{\\alpha}, Q _{\\hat \\alpha} \\} = {1\\over 2} \\delta_{\\alpha \\hat\\alpha} H + {1\\over 2} \\epsilon_{\\alpha\\hat\\alpha} \\widetilde H\\,, ~~~~ \\nonumber \\\\\n&& [E^+, Q_{\\alpha}] = [E^-, Q_{\\hat \\alpha}] = 0\\,, ~~~~ \n [E^+, Q_{\\alpha}] = -\\delta_{\\alpha\\hat\\alpha} Q_{\\alpha} \\,, ~~~~ \n [E^+, Q_{\\hat\\alpha}] = -\\delta_{\\alpha \\hat\\alpha} Q_{\\hat \\alpha}\\,.\\nonumber \n\\end{eqnarray}\n\nWe define a BRST charge of the form\n\\begin{equation}\\label{newq}\nQ = \\oint \\Big( c^+ E_+ + c^- E_- + \\lambda^{\\alpha} Q_{\\alpha} \n+ \\lambda^{\\hat\\alpha} Q_{\\hat\\alpha} + {\\rm bcc ~~ terms} \\Big),\n\\end{equation}\nwhere we have \"gauged\" the generators of the coset $E_\\pm, Q_\\alpha, Q_{\\hat\\alpha}$. \nBy computing the nilpotency of the BRST charge (for simplicity we have \ncomputed only the simple poles) one gets that \n\\begin{equation}\\label{newpA}\n\\{Q, Q\\} = \\oint \\Big( c^+ c^- + {1\\over 2} \\lambda^{\\alpha} \\delta_{\\alpha \\hat \\alpha} \\bar\\lambda^{\\hat \\alpha} \\Big) H + \\Big({1\\over 2} \n\\lambda^{\\alpha} \\epsilon_{\\alpha \\hat \\alpha} \\bar\\lambda^{\\hat \\alpha} \\Big) \\widetilde H,\n\\end{equation}\n where the non-nilpotentcy terms are proportional to the \n Cartan generators $H$ and $\\widetilde H$. So, in order that the entire BRST charge is nilpotent \n we need that \n \\begin{equation}\n\\Big( c^+ c^- + {1\\over 2} \\lambda^{\\alpha} \\delta_{\\alpha \\hat \\alpha} \\bar\\lambda^{\\hat \\alpha} \\Big) =0\\,, \n~~~~~\n\\Big(\\lambda^{\\alpha} \\epsilon_{\\alpha \\hat \\alpha} \\bar\\lambda^{\\hat \\alpha} \\Big) =0\\,.\n\\end{equation}\nTo solve the second one, we can set $\\bar\\lambda^{\\hat\\alpha} = \\lambda^{\\alpha}$ and \nthe first one becomes the supercone constraint (\\ref{coneA}) after a change of coordinates $x = \\lambda^1 + i \\lambda^2$ and $y = \\lambda^1 - i \\lambda^2$. \n\nA detailed discussion on the interpretation of the BRST charge (\\ref{newq}) can be found in \n\\cite{Grassi:2004cz}. The cohomology, the interpretation of the present model, is completely \nopen and it deserves some study. \n\n\\subsection{$Y$-formalism for superprojective spaces} \\par\n\nOne of motivations behind this study is \nthat this new model might be a prototype for supertwistors where not only\nvery limited class of amplitudes are evaluated but also quantum algebra is still unknown. We construct $Y$-formalism for this class of models in\nthis section and wish to clarify these problems in the future work.\n\nThe constraint to which we turn our attention takes the form of a\nsuper-cone (\\ref{coneA}). For simplicity and generalization to other cases,\nit is convenient to rewrite the super-cone constraint (\\ref{coneA}) as\n\\begin{eqnarray}\n\\Phi(\\gamma^i, \\theta^a) \\equiv \\gamma^i g_{ij} \\gamma^j \n+ \\theta^a h_{ab} \\theta^b = 0,\n\\label{constraintA}\n\\end{eqnarray}\nwhere $\\gamma^{i}$ ($\\theta^{a}$) are commuting (anticommuting) fields and \n$i = 1, \\cdots, 2N_{1}$, $a = 1, \\cdots. 2N_{2}$ and\n\\begin{eqnarray}\ng_{i,N_{1}+i} &=& g_{N_{1}+i, i} = \\frac{1}{2}, \\nonumber\\\\\nh_{i,N_{2}+ i} &=& - h_{N_{2}+i,i} = \\frac{1}{2} i,\n\\label{metric}\n\\end{eqnarray}\nand otherwise are vanishing. We shall present the calculations only for the case\n\\begin{eqnarray}\nN_{1} = N_{2} = N.\n\\end{eqnarray}\nThe extension to the general case is straightforward. \n\nLet us start with the action:\n\\begin{eqnarray}\nS = \\int ( \\beta_i \\bar \\partial \\gamma^i - p_a \\bar \\partial \\theta^a).\n\\label{actionA}\n\\end{eqnarray}\nThis action is invariant under the gauge transformations\n\\begin{eqnarray}\n\\delta \\beta_i &=& \\lambda \\gamma_i, \\nonumber\\\\\n\\delta p_a &=& - \\lambda \\theta_a,\n\\label{gaugeA1}\n\\end{eqnarray}\nwhere we have defined the variables with lower indices as\n\\begin{eqnarray}\n\\gamma_i &=& \\gamma^j g_{ji}, \\nonumber\\\\\n\\theta_a &=& \\theta^b h_{ba}.\n\\label{index}\n\\end{eqnarray}\n\nNow let us construct classically gauge-invariant currents whose expressions\nare given as follows:\n\\\\\na) \\underline{Ghost current} \\par\n\\begin{eqnarray}\nJ^0 = \\beta_i \\gamma^i - p_a \\theta^a.\n\\label{ghost current}\n\\end{eqnarray}\nb) \\underline{Stress-energy tensor} \\par\n\\begin{eqnarray}\nT^0 = \\beta_i \\partial \\gamma^i - p_a \\partial \\theta^a.\n\\label{stress}\n\\end{eqnarray}\nc) \\underline{'Lorentz' current} \\par\n\\begin{eqnarray}\nJ_{ij}^0 = \\beta_{[i} \\gamma_{j]}.\n\\label{Lorentz}\n\\end{eqnarray}\nd) \\underline{'Lorentz' current in superspace} \\par\n\\begin{eqnarray}\nj_{ab}^0 = p_{(a} \\theta_{b)}.\n\\label{superLorentz}\n\\end{eqnarray}\n\nNext, let us construct the $Y$-formalism. Firstly, let us introduce\ntwo kinds of $Y$-fields by\n\\begin{eqnarray}\nY_i &=& \\frac{v_i}{v_k \\gamma^k + w_c \\theta^c}, \\nonumber\\\\\nY_a &=& \\frac{w_a}{v_k \\gamma^k + w_c \\theta^c}.\n\\label{Y}\n\\end{eqnarray}\nThe constant vectors $v_i$ and $w_a$ are commuting and anticommuting numbers, respecively. \n\\footnote{For simplicity, we have used the additional constraints\n$v_i v^i + w_a w^a = 0$ or equivalently, in the $Y$-fields, we have \n$Y_i Y^i + Y_a Y^a = 0$. They are not essential, but they simplify the computations. \n}\nMoreover, the above definition of $Y$-fields leads to the following\nuseful relations:\n\\begin{eqnarray}\n1 &=& Y_i \\gamma^i + Y_a \\theta^a, \n\\nonumber\\\\\n\\partial Y_i &=& - (Y_k \\partial \\gamma^k + Y_c \\partial \\theta^c) Y_i,\n\\nonumber\\\\\n\\partial Y_a &=& - (Y_k \\partial \\gamma^k + Y_c \\partial \\theta^c) Y_a,\n\\label{relations}\n\\end{eqnarray}\nwhich will be often utilized in calculating an algebra below.\n\nSecondly, we make the following projection operators:\n\\begin{eqnarray}\nK_i \\ ^j &=& Y^j \\gamma_i = \\frac{v^j \\gamma_i}{v_k \\gamma^k + w_c \\theta^c}, \\nonumber\\\\\nK_i \\ ^a &=& Y^a \\gamma_i = \\frac{w^a \\gamma_i}{v_k \\gamma^k + w_c \\theta^c}, \n\\nonumber\\\\\n\\nonumber\\\\\nK_a \\ ^b &=& Y^b \\theta_a = \\frac{w^b \\theta_a}{v_k \\gamma^k + w_c \\theta^c},\n\\nonumber\\\\\nK_a \\ ^i &=& Y^i \\theta_a = \\frac{v^i \\theta_a}{v_k \\gamma^k + w_c \\theta^c}. \n\\label{projection}\n\\end{eqnarray}\nUsing these definition of projection operators together with the super-cone\nconstraint and the relations (\\ref{relations}), we can derive the \nfollowing useful equations: \n\\begin{eqnarray}\nK_i \\ ^i - K_a \\ ^a &=& K_i \\ ^j K_j \\ ^i - K_a \\ ^b K_b \\ ^a\n- 2 K_i \\ ^a K_a \\ ^i = 1, \n\\nonumber\\\\\nK_j \\ ^i \\gamma^j + K_b \\ ^i \\theta^b &=& K_j \\ ^a \\gamma^j + K_b \\ ^a \\theta^b = 0, \n\\nonumber\\\\\nK_j \\ ^i \\partial \\gamma^j + K_b \\ ^i \\partial \\theta^b \n&=& K_j \\ ^a \\partial \\gamma^j + K_b \\ ^a \\partial \\theta^b = 0, \n\\nonumber\\\\\nY_j K_i \\ ^j + Y_b K_i \\ ^b &=& Y_j K_a \\ ^j + Y_b K_a \\ ^b = 0, \n\\nonumber\\\\\nY_j \\partial K_i \\ ^j + Y_b \\partial K_i \\ ^b \n&=& Y_j \\partial K_a \\ ^j + Y_b \\partial K_a \\ ^b = 0, \n\\label{relations2}\n\\end{eqnarray}\nwhich will be also used in calculating an algebra.\n\nThirdly, we set up the basic OPE's\n\\begin{eqnarray}\n< \\beta_i(y) \\gamma^j(z) > &=& \\frac{1}{y-z} (\\delta_i^j - K_i \\ ^j(z)), \\nonumber\\\\\n< \\beta_i(y) \\theta^a(z) > &=& - \\frac{1}{y-z} K_i \\ ^a(z), \n\\nonumber\\\\\n< p_a(y) \\theta^b(z) > &=& \\frac{1}{y-z} (\\delta_a^b - K_a \\ ^b(z)),\n\\nonumber\\\\\n< p_a(y) \\gamma^i(z) > &=& \\frac{1}{y-z} K_a \\ ^i(z).\n\\label{OPE}\n\\end{eqnarray}\nThen, it is easy to show that $< \\beta_i(y) \\Phi(z) > = < p_a(y) \\Phi(z) >\n= 0$.\n\nNow we move on to a derivation of an algebra among gauge-invariant quantum currents.\nRequiring that terms depending on $Y$-fields, which violate the Lorentz symmetry,\nshould be absent in the algebra, we can fix the expression of gauge-invariant quantum currents uniquely. Actually, the currents read\n\\\\\na) \\underline{Ghost current} \\par\n\\begin{eqnarray}\nJ = \\beta_i \\gamma^i - p_a \\theta^a + \\frac{3}{2} (Y_i \\partial \\gamma^i\n+ Y_a \\partial \\theta^a).\n\\label{q-ghost current}\n\\end{eqnarray}\nb) \\underline{Stress-energy tensor} \\par\n\\begin{eqnarray}\nT = \\beta_i \\partial \\gamma^i - p_a \\partial \\theta^a \n+ \\frac{1}{2} \\partial (Y_i \\partial \\gamma^i + Y_a \\partial \\theta^a).\n\\label{q-stress}\n\\end{eqnarray}\nc) \\underline{'Lorentz' current} \\par\n\\begin{eqnarray}\nJ_{ij} = \\beta_{[i} \\gamma_{j]} - Y_{[i} \\partial \\gamma_{j]}\n- \\frac{1}{2} \\partial Y_{[i} \\gamma_{j]}.\n\\label{q-Lorentz}\n\\end{eqnarray}\nd) \\underline{'Lorentz' current in superspace} \\par\n\\begin{eqnarray}\nj_{ab} = p_{(a} \\theta_{b)} + Y_{(a} \\partial \\theta_{b)}\n+ \\frac{1}{2} \\partial Y_{(a} \\theta_{b)}.\n\\label{q-superLorentz}\n\\end{eqnarray}\n\nIt then turns out that the algebra among currents is of form\n\\begin{eqnarray}\n &=& \\frac{1}{2 (y-z)^2} \n(g_{k[i} g_{j]l} - g_{l[i} g_{j]k}) \n+ \\frac{1}{y-z} (g_{k[i} J_{j]l} - g_{l[i} J_{j]k}),\n\\nonumber\\\\\n &=& \\frac{1}{2 (y-z)^2} \n(h_{c(a} h_{b)d} + h_{d(a} h_{b)c}) \n+ \\frac{1}{y-z} (h_{c(a} j_{b)d} + h_{d(a} j_{b)c}),\n\\nonumber\\\\\n &=& 0,\n\\nonumber\\\\\n &=& 0,\n\\nonumber\\\\\n &=& \\frac{1}{(y-z)^2} J_{ij}(y),\n\\nonumber\\\\\n &=& \\frac{1}{(y-z)^2} j_{ab}(y),\n\\nonumber\\\\\n &=& \\frac{-1}{(y-z)^4} + \\frac{1}{(y-z)^2}\n(T(y) + T(z)),\n\\nonumber\\\\\n &=& \\frac{-2}{(y-z)^3} + \\frac{1}{(y-z)^2} J(y),\n\\nonumber\\\\\n &=& \\frac{4}{(y-z)^2}. \n\\label{algebra2}\n\\end{eqnarray}\nCompared this algebra with that of the bosonic $\\beta-\\gamma$ system, we find that this system corresponds to the case $N = 0$. This fact can be easily\nunderstood since the dynamical degrees of freedom are canceled between\nbosons and fermions.\n\n\\section*{Acknowledgments} \nThe work of P.A.G. and of M.T. is supported in part by the European Community's Human Potential Programme under contract MRTN-CT-2004-005104, \"Constituents, Fundamental Forces and Symmetries of the Universe\". The work of M.T. is also supported \nby INTAS 05-10000087928. P.A.G. would like to thank C. Imbimbo, S. Giusto and G. Policastro for useful discussions. I.O. is grateful to T. Tokunaga for\nvaluable discussions.\n\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Appendix}\n\n\\section{Conclusion} \n\\label{sec:conclusion}\n\nIn this paper, we developed a system design that supports a large-scale oblivious search on unspent transaction outputs for Bitcoin SPV clients while efficiently maintains the state of the Bitcoin $\\mathsf{UTXO}$ set via an oblivious update protocol. \nOur design leverages the TEE capabilities of Intel SGX to provide strong privacy and security guarantees to Bitcoin SPV client even with the presence of a potentially malicious server.\nMoreover, by putting reasonable assumptions on the accessing frequency of the SPV clients, we present novel ORAM construction that offers both privacy and efficiency to the clients.\nWe showed that the prototype of the system is much more efficient than the use of standard ORAM construction as it is. \nIn particular, due to the use of two ORAM trees in the design of \\mbox{$T^3$}\\xspace, we improve the performance of an ORAM access by two time and allow the system to handle concurrent client's requests.\nAlso, our implementation shows one order of magnitude performance gain when combining recursive ORAM construction the current existing construction to stress the importance of using recursive ORAM construction in TEE with restricted memory.\nFinally, while the applicability of $T^3$\\xspace in cryptocurrencies beyond Bitcoin is apparent, we believe our work will motivate further research on oblivious memory with the restricted access patterns.\n\n\\section{Proposed System}\n\\label{sec:new-protocol}\n\nIn this section, we first describe how $T^3$\\xspace stores the UTXO set by exploring different mappings between the unspent transaction outputs and the ORAM blocks.\nWe see that naive mapping may not be secure for blockchain applications as it may lead to denial of services attacks. \nNext, we demonstrate how Intel SGX can be considered as a trusted execution unit to access ORAM and perform read\/write operations in an oblivious manner. \nFinally, we will describe how the system handles clients' requests during a write operation.\n\n\\subsection{Storage Structure of the UTXO set}\n\\label{subsec: Oblivious Storage of the UTXO set}\nIn the first step, we show how to map the public key hash to ORAM block identification and then describe the storage requirements in $T^3$\\xspace. \n\n\\subsubsection{Bitcoin unspent transaction output mapping}\n\\label{subsubsec: btcintoORAM}\n\nIn the design of \\mbox{$T^3$}\\xspace, we assume that the SPV clients only know his\/her addresses (i.e., the public key hashes); \ntherefore, to return the outputs belonging to the client's address, the enclave needs to know the mapping between the address and the ORAM block identification. \n\n\nIn this work, we propose two simple mappings to store unspent outputs in the ORAM tree. \nMore specifically, both approaches use standard cryptographic hash functions along with a secret key generated by the enclave.\nThe first approach is to map a single Bitcoin address into a single ORAM block, \nand the second approach is to map a Bitcoin address into multiple ORAM blocks. \nWe will later explain the trade-off between these two approaches. \nIntuitively, the first approach is more efficient in terms of performance and can be more expensive in terms of storage overhead. \nThe second approach gives some flexibility in terms of storage overhead; however, to offer strong privacy to every address, this approach can be more expensive in terms of performance because it may incur more ORAM calls. \n\n\\noindent\n\\paragraph{Single address into Single ORAM block.}\n\\label{par:key-hash-with-oram}\nIn this design, during the initialization, we require the program inside the enclave to use a keyed hash function to map the public key hash to ORAM block identification. \nThe secret key of the hash function is generated and known only by the enclave.\nIn other words, the mapping between a Bitcoin address to an ORAM block identification is known only to the SGX.\nWe define the mapping as follow:\n\\begin{compactitem}\n\t\\item $\\mathsf{bid}\\leftarrow \\mathsf{OBlockMap}({pkh}, k_b)$: the function takes as input a $20$-bytes hash digest ${pkh}$ and a secret key $k_b$, it outputs the block identification number $\\mathsf{bid}\\in \\{0,\\dots,N-1\\}$. \n\\end{compactitem}\n\nThe key-hashing approach offers some flexibility when deciding the size of an ORAM blocks and the size of height of the ORAM tree. \nThese two factors affect the size of the position map (resp. number of recursive levels) for non-recursive (resp. recursive) ORAM constructions. \t\nHowever, since the output domain of $\\mathsf{OBlockMap}(\\cdot, \\cdot)$ is limited to the size of the ORAM blocks, there will exist collisions. \nThe following claim gives us a loose upper bound on the number of addresses that should be stored inside an ORAM block.\n\\begin{claim}(Addresses per ORAM block)\\label{claim:addressesperoramblock}\n\tLet $m$ be the number of public key hashes, $N$ be the number of ORAM blocks. If the $\\mathsf{OBlockMap}()$ acts as a truly random function, then the maximum number of addresses in each ORAM block is smaller than $e\\cdot m\/N$ with a probability $1-1\/N$.\n\\end{claim}\n\\begin{proof}\n\tThis is a standard max-load analysis. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\tWe refer readers to \\cite{czumaj-bin-ball} for detailed analysis.\n\tWe note that there exists a tighter bound, but we use $e\\cdot m\/N$ bounds to simplify the equation. \n\\end{proof}\n\nThus, if we limit each ORAM block to contain the outputs of at most $e\\cdot m\/N$ addresses, then the probability that every address is included is at least $1-1\/N$. \\Cref{fig:single-add-to-single-block} gives us a high level overview of this approach.\n\\begin{figure}[h!]\n\t\\centering\n\t\\resizebox{.50\\linewidth}{!}{\n\t\\input{one-to-one-map}\n\t}\n\t\\caption{Single address into Single ORAM block}\n\t\\label{fig:single-add-to-single-block}\n\\end{figure}\n\n\n\n\n\\noindent\\textbf{Single address into Many ORAM blocks.}\nMapping a single address into a single ORAM block incurs less work on the server as it requires a single ORAM access for an address.\nHowever, if we want to allow each address to have more than one output, \nusing the first approach implies that the storage overhead increase linearly \nbecause the first approach distribute unspent outputs based on its addresses. \nTherefore, we have to pad dummy data for those addresses that contain $1$ outputs. \nThus, we need a different mapping without linear increasing in storage overhead. \nTo fix this shortcoming, the system needs to assign unspent outputs into ORAM block uniformly. \nOne method is to allow a client to specify the number of ORAM accesses to obtain all of its unspent outputs as long as the number of requests does not exceed certain threshold. \nWe define the mapping as follows:\n\\begin{compactitem}\n\t\\item $ \\{bid_i\\}_{i \\in \\{0,\\dots, \\delta-1 \\}} \\leftarrow \\mathsf{OBlockMap}(pkh, k_b, \\delta)$: the function takes as input a $20$-bytes hash digest ${pkh}$, a secret key $k_b$, and a number $\\delta$ where the maximum value of $\\delta$ is specified by the system. It outputs a set of block identification numbers $ \\{bid_i\\}_{i \\in \\{0,\\dots, \\delta-1 \\}} \\subseteq \\{0,\\dots,N-1\\}$. \n\\end{compactitem}\nThis approach also introduces some leakage as some addresses may contain more unspent outputs than others. \nAlternatively, the system can fix the value of $\\delta$ ORAM accesses for all addresses with the expense of performance (i.e., one address incurs constant ORAM accesses). Similarly, the storage overhead of \\mbox{$T^3$}\\xspace can be computed using the following claim:\n\\begin{claim}(UTXO per ORAM block)\\label{claim:utxoperoramblock}\n\tLet $m$ be the number of unspent outputs, $N$ be the number of ORAM blocks. If the $\\mathsf{OBlockMap}$ acts as a truly random function, then the maximum number of outputs in each ORAM block is smaller than $e\\cdot m\/N$ with probability at least $1 - 1\/N$\n\\end{claim}\nThe proof is identical to proof of claim~\\ref{claim:addressesperoramblock}. \n\nFigure~\\ref{fig:single-add-to-many-block-fixed-not-fixed} offers an overview of the both approaches.\n\\begin{figure}[h]\n\t\\centering\n\t\\resizebox{\\linewidth}{!}{\n\t\\input{one-to-many-3}\n\t}\n\n\n\n\t\\caption{Single Address into Many ORAM blocks}\n\t\\label{fig:single-add-to-many-block-fixed-not-fixed}\n\\end{figure}\n\\subsubsection{Storage}\\\nIn this system, we require the untrusted server to store three separate databases which are the \\textit{read-once} ORAM tree\\xspace, the \\textit{original} ORAM tree\\xspace, and the blockheader chain. In particular,\n\t\\textbf{\\textit{Read-Once} ORAM Tree} \n\n\tserves as a dedicated storage to handle clients' requests. \n\n\tThe structure of the tree is identical to the standard ORAM tree\n\n\n\n\n\t\\textbf{\\textit{Original} ORAM Tree} is where all standard ORAM eviction operations are performed. \n\n\n\n\n\n\n\tIn this work, we also require the enclave to maintain the \\textbf{Bitcoin Header Chain} to verify the proof of work of the bitcoin block sent by other bitcoin client. The header chain is stored in the untrusted memory with integrity check.\n\\subsection{Oblivious Read and Write Protocols}\n\\label{sub:oRAM operations}\nIn $T^3$\\xspace, the SPV client is the party who invokes read accesses, and the Bitcoin network is the party who invokes write accesses. \nThe TEE in the server is the one that performs both of those accesses on behalf of the client and the Bitcoin network. \n\\subsubsection{Server System Components}\nBefore explaining how oblivious read and write accesses work, we first start outlining the different components of our design.\nThe server is initialized with different enclaves:\n\\paragraph{Managing Enclave $\\mathcal E_{m}$}\ncoordinates other enclaves and to handle requests from the clients. \nThe \\textit{managing} enclave also handles the communication with other Bitcoin client or local Bitcoin client ($\\mathsf{bitcoind}$) via request procedure calls (RPC) to obtain Bitcoin blocks. \nUpon receiving the Bitcoin block, the \\textit{managing} enclave\\xspace also verifies the integrity of the block using a separated header chain.\n\\paragraph{Reading Enclave $\\mathcal E_{r}$} is a dedicated enclave initialized by the \\textit{managing} enclave\\xspace. \nIt has a copy of ORAM position map and its own stash. The \\textit{reading} enclave\\xspace operates on the \\textit{read-once} ORAM tree\\xspace. \nAlso, the \\textit{reading} enclave\\xspace only performs ORAM $\\mathsf{ReadPath}$ operations to obtain data while ORAM $\\mathsf{Eviction}$ operations will be handled by the \\textit{writing} enclave\\xspace.\n\\paragraph{Writing Enclave $\\mathcal E_{w}$} performs $\\mathsf{Eviction}$ procedure for each read request, and performs ORAM writing accesses when a new Bitcoin block arrives from the Bitcoin network.\n\n\\subsubsection{Oblivious \\textit{\\textit{read-once}}\\xspace Protocol}\n\\label{subsub:read-proc}\nHere, we describe how a remote client can perform a read access on the UTXO set. \n\n\\paragraph{Notation. } \nFirst, let's denote \n$K_b$ to be the block mapping key, \n$\\mathsf{bid}$ to be the ORAM block identification. \nWe let $(\\mathsf{Enc}, \\mathsf{Dec})$ denote an authenticated encryption scheme.\nWe assume that the the server has already been initialized with a \\textit{writing} enclave\\xspace, $\\mathcal E_{w}$\\xspace and a \\textit{managing} enclave\\xspace, $\\mathcal E_{m}$\\xspace. \nThe \\textit{managing} enclave\\xspace has a similar copy of the position map as the map in the \\textit{writing} enclave\\xspace.\nFigure~\\ref{fig:read-overview} presents the oblivious read protocol of $T^3$\\xspace. The oblivious read protocol can be described as follows:\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.95\\columnwidth]{fig\/read_overview.pdf}\n\\caption{The read protocol.\nSteps \\protect\\bcircled{1}-\\protect\\bcircled{5} describes how \\protect\\mbox{$T^3$}\\xspace receives and responds to the client, and for each request, the \\textit{writing} enclave\\xspace performs the $\\mathsf{Eviction}$ procedure of ORAM on the \\textit{original} ORAM tree\\xspace during step \\protect\\bcircled{6}.\n}\n\\label{fig:read-overview}\n\\end{figure}\n\\begin{asparaenum}\n\t\\item \\textbf{The client establishes a secure channel with the \\textit{managing} enclave\\xspace}~\\bcircled{1}: First, the client performs a remote attestation with the secure \\textit{managing} enclave, $\\mathcal E_{m}$\\xspace , and agrees on a session key, $K_s$. \n\tThe client encrypts his address along with the proof of ownership of that address, and sends the encrypted query to the server to be passed to $\\mathcal E_{m}$\\xspace. For simplicity, we assume that the plaintext only contains a public key hash, $pkh$, that the client is interested in, and the proof of ownership of the $pkh$ is $\\phi$, $C\\leftarrow\\mathsf{Enc}_{K_s}(pkh, \\phi)$. \n\tNote that there are different ways to prove the ownership of public key hash\/addresses. \n\tIn Bitcoin, if the public key is never revealed before, the proof of ownership can simply be the public key (i.e. $\\phi = pk$ such that $H(pk)=pkh$).\n\tAlternatively, the system can enforce a client to provide the signature and the public key to prove the ownership of the public key hash.\n\n\t\\item \\textbf{The \\textit{managing} enclave initializes a \\textit{reading} enclave}~\\bcircled{2}~: \n\tafter receiving a client's request, $\\mathcal E_{m}$\\xspace initializes a dedicated \\textit{reading} enclave\\xspace, $\\mathcal E_{r}$\\xspace to handle the client's future requests. \n\tAlso, we require that the enclaves authenticate each other, and the existence of a secure channel between enclaves. \n\tMoreover, the \\textit{reading} enclave\\xspace has its copy of the position map, its own stash, the block mapping key $K_b$, and the agreed session key $K_s$.\n\n\t\\item \\textbf{The \\textit{managing} enclave identifies and forwards ORAM Block ID to both \\textit{reading} and \\textit{writing} enclaves}~\\bcircled{2}: \n\tAfter decrypting the ciphertext $(pkh, \\phi) \\leftarrow \\mathsf{Dec}_{K_s}(C)$, $\\mathcal E_{m}$\\xspace verifies the proof $\\phi$ and $pkh$,\n\tthen uses $\\mathsf{OBlockMap}(\\cdot,\\cdot)$~\\footnote{for simplicity, we assume that the one-to-one mapping is used here} function to learn the ORAM block ID, $\\mathsf{bid} \\leftarrow \\mathsf{OBlockMap}(pkh, K_b)$ where $K_b$ is the secret key generated by the enclave during initialization for mapping purposes. \n\tAfter obtaining the ORAM id, $\\mathsf{bid}$, the \\textit{managing} enclave forwards $\\mathsf{bid}$ to the \\textit{writing} enclave\\xspace for the eviction procedure, \n\tand forwards the $(pkh,\\mathsf{bid})$ to the \\textit{reading} enclave\\xspace.\n\n\t\\item \\textbf{The \\textit{reading} enclave\\xspace performs \\textit{read-once} ORAM access\\xspace on the \\textit{read-once} ORAM tree\\xspace}~\\bcircled{3}: Based on the given $\\mathsf{bid}$, the \\textit{reading} enclave\\xspace performs ORAM read only accesses on the ORAM tree to obtain the block. \n\tIf the block contains the unspent output that belongs to the public key $pkh$, \n the \\textit{reading} enclave\\xspace adds outputs into the response $R$. \n\tTo mitigate the size leakage, the response $R$ is padded with dummy data if there is no UTXO found.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\\item \\textbf{The \\textit{reading} enclave\\xspace responds to the Client}~\\bcircled{4}-\\bcircled{5}~: The enclave encrypts the response, $R$, using the session key $K_s$ then sends it to the client. \n\t\\item \\textbf{The \\textit{writing} enclave\\xspace performs the eviction procedure on the \\textit{original} ORAM tree\\xspace}~\\bcircled{6}: After obtaining the $\\mathsf{bid}$ from the \\textit{managing} enclave\\xspace, the update enclave will perform a standard ORAM read accesses on the \\textit{original} ORAM tree\\xspace. \n\tThe goal of this procedure is to use the $\\mathsf{Eviction}$~procedure inside standard ORAM operation to rerandomize the location of the actual block. No actual data is return in this step.\n\t\\end{asparaenum}\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Oblivious Write Protocol}\n\\label{subsub:write-proc}\nIn the Bitcoin network, miners generate a new Bitcoin block on average every 10 minutes. \nWhen the server receives a new block from the Bitcoin network, the \\textit{managing} enclave\\xspace can obtain it from the bitcoin client. \nThe $\\mathcal E_{m}$\\xspace verifies the integrity of the block by computing the Merkle root from transactions, then verifying the proof of work, \nand the \\textit{writing} enclave\\xspace has to perform an update on the \\textit{original} ORAM tree\\xspace; \nhowever, in the mean time, the system should be able to handle clients' requests during updates.\nWe will explain how $T^3$\\xspace handles oblivious write accesses while handling clients' requests as follow:\n\n\n\n\n\\begin{asparaenum}\n\t\\item \\textbf{The \\textit{managing} enclave verifies a new Bitcoin block}~\\rcircled{1}-\\rcircled{3}~: Once a bitcoin block arrives to the system from the Bitcoin network, the \\textit{managing} enclave\\xspace $\\mathcal E_{m}$\\xspace can obtain it from the Bitcoin client. \n\tHowever, since the client runs outside the enclave, the enclave needs to verify the integrity of the new block by computing the Merkle root and verifying the proof of work to make sure that the block has not been tampered by the untrusted OS. \n\tFor the detail of these computations, we refer readers to \\cite{btc-reference}. \n\tMoreover, as discussed in section~\\ref{subsec: Oblivious Storage of the UTXO set}, to verify a newly arrived block, the system is required to keep a separate block headers chain with integrity check in the untrusted memory. \n\tOnce $\\mathcal E_{m}$\\xspace verifies the bitcoin block, $\\mathcal E_{m}$\\xspace starts pruning the transactions to obtain relevant information of the transactions' inputs and outputs. \n\tThen, $\\mathcal E_{m}$\\xspace uses $\\mathsf{OBlockMap}(\\cdot, \\cdot)$ to find the ORAM block identification to queue up ORAM write requests to the \\textit{writing} enclave\\xspace. \n\tDuring this process, the oblivious read protocol performs as normal on the \\textit{read-once} ORAM tree\\xspace. \n\n\\begin{figure}[t]\n\t\\centering\n\n\t\\includegraphics[width=.95\\columnwidth]{fig\/write_overview.pdf}\n\t\\caption{Oblivious write protocol. \n\tDuring steps \\protect\\rcircled{1}-\\protect\\rcircled{5}, the \\textit{managing} enclave\\xspace receives and responds to SPV client request as usual. During steps \\protect\\rcircled{6}-\\protect\\rcircled{8}, read requests from clients are queued up, and the \\textit{managing} enclave\\xspace resume these requests after updating the \\textit{read-once} ORAM tree\\xspace\n\t}\n\t\\label{fig:write-overview}\n\\end{figure}\n\n\t\\item \\textbf{The \\textit{managing} enclave sends write requests to the \\textit{writing} enclave\\xspace}~\\rcircled{4}: \n\t\tOnce the pruning process completes, the $\\mathcal E_{m}$\\xspace starts sending write requests based on data extracted from the bitcoin block to the \\textit{writing} enclave\\xspace, $\\mathcal E_{w}$\\xspace. \n\t\tOn otherhand, for each eviction request resulted from SPV client's requests, $\\mathcal E_{m}$\\xspace starts queuing up those eviction requests. \n\n\t\\item \\textbf{The \\textit{writing} enclave performs write accesses on the \\textit{original} ORAM tree\\xspace}~\\rcircled{4}-\\rcircled{5}: \n\t\tUpon receiving writing requests from $\\mathcal E_{m}$\\xspace, the $\\mathcal E_{w}$\\xspace performs all writing requests in the writing queue on the \\textit{original} ORAM tree\\xspace. \n\n\t\\item \\textbf{The \\textit{writing} enclave\\xspace finishes all eviction requests queued up on the \\textit{original} ORAM tree\\xspace}~\\rcircled{6}-\\rcircled{7}:\n\t\tOnce finished updating the tree, the $\\mathcal E_{w}$\\xspace signals $\\mathcal E_{m}$\\xspace to start queuing up clients' requests and performs all eviction requests incurred by SPV clients' read requests during update interval.\n\t\tFinally, when it finishes, it signals the $\\mathcal E_{m}$\\xspace to update the \\textit{read-once} ORAM tree\\xspace and make a copy of the position map. \n\t\n\t\\item \\textbf{The \\textit{managing} enclave performs an update the \\textit{read-once} ORAM tree\\xspace and the \\textit{original} ORAM tree\\xspace and enclave metadata}~\\rcircled{8}: In particular, $\\mathcal E_{m}$\\xspace discards the current copy of the \\textit{read-once} ORAM tree\\xspace, and makes 2 identical copies of the most updated \\textit{original} ORAM tree\\xspace. One is used as \\textit{read-once} ORAM tree\\xspace, and the other is used as \\textit{original} ORAM tree\\xspace. Also, the new position map and new stash are updated for the \\textit{managing} enclave\\xspace. \n\tOnce this process is finished, $\\mathcal E_{m}$\\xspace starts answering SPV clients' requests again.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{asparaenum}\nFigure~\\ref{fig:write-overview} gives us an overview of the oblivious write protocol.\n\n\\paragraph{Discussion}\nThe core idea of the oblivious update protocol is to minimize the downtime of $T^3$\\xspace during the update process. More specifically, during step~\\rcircled{1}-\\rcircled{5}, $T^3$\\xspace still allows SPV clients to query the system while from step~\\rcircled{6}-\\rcircled{8}, $T^3$\\xspace stops accepting clients' requests in order to synchronize both trees. This approach introduces some delay; however, the system downtime is minimized to the same amount of time it takes for the \\textit{writing} enclave\\xspace to finish all eviction requests. \n\n\n\n\n\n\n\n\n\\section{Design Rationale}\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nOver the last few years, we have seen a great interest in public blockchain in the community. \nThe Bitcoin blockchain offered a way to provide security and privacy for financial transactions. \nHowever, due to the huge adoption by the community, the size of the Bitcoin blockchain has become too large for small and resource-constrained devices such as personal laptops or mobile phones, raising not only performance but also privacy concerns in the community.\nAs of October 2018, the size of the unindexed Bitcoin blockchain is 230 GB.\n\nTo this end, Bitcoin's simplified payment verification (SPV) client has become a widely-adopted solution to resolve a storage problem for constrained devices. \nNakamoto~\\cite{Nakamoto_bitcoin:a} sketched the idea of SPV clients in the Bitcoin whitepaper, \nand in the Bitcoin improvement proposal 37 (BIP37)~\\cite{BIP37}, Mike Hearn combines Nakamoto's idea with the use of Bloom filters to standardize the design of Bitcoin SPV clients. \nThis design has become de facto standard and been used by other SPV clients such as BitcoinJ~\\cite{bitcoinj-cite} and Electrum~\\cite{electrum-cite}.\n\nThe core of SPV clients is in only downloading and then verifying part of the blockchain that is relevant to the SPV client itself.\nIn particular, the SPV client loads its addresses into a Bloom filter and sends the filter to a Bitcoin full client, and \nThe Bitcoin full client will use that filter to identify if a block contains transactions that are relevant to the SPV client, \nand once it finds the block, it will send a modified block that only contains relevant transactions along with Merkle proofs for those transactions. \n\nHowever, the current SPV solution relied on Bloom filters raises security and privacy concerns to the SPV clients when communicates with potentially malicious nodes.\nIn particular, Gervais et al.~\\cite{Gervais:2014:SPV-privacy} show that it is possible for a malicious node to learn several addresses of the client from the Bloom filter with high probability.\nMoreover, if the adversarial node can collect two filters issued by the same client, then a considerable number of addresses owned by the client will be leaked. \n\nTo provide a strong privacy guarantee for SPV clients, one needs a solution that can hide wallets\/addresses queried by the SPV clients. \nWhile such a system can be built using private information retrieval (PIR) primitive, the existing cryptographic PIR solutions~\\cite{JaschkeGAS17} are not been practical to scale to handle millions of Bitcoin users.\nOn the other hand, to gain more efficiency, one can use ORAM and trusted execution environment to propose generic PIR systems~\\cite{thang-hoang:posup-popets,oblidb,SasyGF18-zero-trace}. \nHowever, as it becomes apparent in the later in this paper, naively combining ORAM scheme as it is with TEE makes \nthe practicality of those generic systems questionable when used in a large network like Bitcoin due to the lack of concurrency in ORAM as well as the limitation of TEE with restricted memory.\n\n\n\n\\noindent\\textbf{Our Contribution.} \nThis work aims not only to design a system that provides SPV clients with privacy-preserving access to the Bitcoin blockchain data but also to consider other practical aspects on how to scale such a system to handle client requests in a large-scale.\nOur contributions can be summarized as follows:\n\nFirstly, we present a novel design for a system that can handle up to thousands of requests per minute from Bitcoin SPV clients based on a \n\\textit{restricted access} Oblivious Random Access Memory (ORAM) and the trusted execution capabilities of TEE. \nIn particular, one of the main contributions of our design is the optimization access in the prominent tree-based ORAM schemes\nthat allow those ORAM schemes to support concurrent accesses which is essential for handling SPV clients' requests. \nIn this design, the access privacy guarantee is still maintained because of our natural assumption that the rational Bitcoin SPV clients should only query for their particular transaction {\\em once} before the arrival of a new Bitcoin block. \nNevertheless, we later show that even when the SPV clients are irrational then the privacy for such clients is only compromised for a short period of time. \nThe security guarantee of \\mbox{$T^3$}\\xspace also relies on the trusted execution capabilities of TEE that allows SPV clients to perform ORAM operations securely and remotely.\nOur generic design works with other blockchains, any tree-based ORAM schemes~\\cite{Elaine-rORAM,Stefanov:2013,wang-circuit-oram-2015}, and any TEE with attestation capability.\n\n\n\n\nSecondly, we implemented a prototype of \\mbox{$T^3$}\\xspace and evaluated its performance to demonstrate the practicality of our approach. \nMore specifically, we extracted the unspent transaction outputs set of Bitcoin in October 2018 and used it to measure the performance of the system when handling clients' requests. \nThe implementation of \\mbox{$T^3$}\\xspace also adopts standard techniques (i.e., oblivious operations using $\\mathsf{cmov}$~\\cite{SasyGF18-zero-trace,ndss-AhmadKSL18,racoon}) to be secure against known side-channel attacks~\\cite{shadow-branch-lee-usenix17,hid-sgx-sidechannel-usenix17,Xu15ControlledChannel}.\nMoreover, the use of recursive ORAM constructions in \\mbox{$T^3$}\\xspace makes the system much more suitable for TEE with restricted trusted memory like Intel SGX. \nWe then show that the running time of the ORAM read access decreases linearly with the number of the threads used (e.g, up to $8\\times$ performance gained with $4$ threads). \n\n\n\n\n\nFinally, we conclude that putting natural restrictions on the access patterns on oblivious memory can lead to significant performance improvement and better ORAM design. While the applicability of $T^3$\\xspace in cryptocurrencies beyond Bitcoin is apparent, we believe our work will also motivate further research on oblivious memory with restricted access patterns.\n\n\\paragraph{Concurrent Work.}\nThe soon-to-be published BITE system~\\cite{bite-spv-sgx} also employs the Oblivious Database construction for SPV client privacy.\nThe main idea of the BITE construction is to combine the use of non-recursive \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace~\\cite{Stefanov:2013} construction and TEE (such as Intel SGX) to propose a generic system that offers SPV client with oblivious access to the database. \nHowever, BITE did not address several shortcomings of using \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace as it is and TEE with restricted memory in practice. \nIn particular, the BITE design did not consider use recursive ORAM constructions to reduce the trusted memory usage; therefore, the efficiency of the system will be degraded once the size of the database gets too large. \nMoreover, due to the inherent lack of concurrency in tree-based ORAM such as \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace, naively using Path-ORAM makes BITE unsuitable for handling thousands of Bitcoin client's requests per minute as well as thousands of updates every fixed period of times (e.g., 10 minutes for Bitcoin).\nIn this work, we investigate the use of both recursive \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace and recursive \\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace to understand the actual performance and the actual storage overhead put on the server. Importantly, we propose a two-tree ORAM design to further enhance the performance of standard ORAM accesses as well as to allow concurrent requests from the SPV client. \n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Evaluation and Comparison}\n\\label{sec:Evaluation}\n\nIn this section, we describe our configuration, our experimental results, and the storage overhead of the system based on the analysis of the UTXO set on the Bitcoin blockchain. Moreover, we give a comparison between \\mbox{$T^3$}\\xspace and the current existing SPV solution in term of performance and communication overhead. Finally, we address the capabilities of \\mbox{$T^3$}\\xspace compared to other related works. \n\\subsection{Configuration}\n\\label{sub:configuration}\n\\paragraph{Software. } \nWe implemented our system with C++ using Intel SGX SDK v2.0.\nThe implementation of the ORAM controller is built on top the $\\mathsf{Zerotrace}$~\\cite{SasyGF18-zero-trace} implementation. In order to handle the communication with the Bitcoin network, \nwe have used \\texttt{libjson-rpc-cpp}~\\cite{libjson-rpc} framework to build C++ wrapper functions to communicate with the Bitcoin daemon (\\texttt{bitcoind}~\\cite{bitcoin-core}) from inside the enclave through JSON-RPC calls. For extracting the UTXO database, we used the \\texttt{bitcoin-tool} implementation proposed in~\\cite{analysis-of-utxo}. \nThis allows us to save time during the initialization phase. Finally, we used \\texttt{python-bitcoinlib}~\\cite{pythonbitcoin} to compare the performance of \\mbox{$T^3$}\\xspace with the current existing SPV solution.\n\n\\paragraph{Database. } To reduce the time of initializing both ORAM trees from the \\textit{genesis} block, we used $\\texttt{bitcoin\\text{-}tool}$ implementation proposed in~\\cite{analysis-of-utxo} to extract $3.2$GB of the Bitcoin {UTXO} set in February 2019. \n\nWe have downloaded a snapshot of the Bitcoin blockchain including block 0 to $551,731$, containing a total of $58,156,895$ Unspent Transaction Outputs (UTXO). \n\\Cref{fig:utxo-analysis} shows the distribution of the unspent transaction outputs per address. \nWe see that more than 90\\% of the addresses have less than three UTXOs. \nIn our prototype, we considered at most two UTXOs per wallet ID. \nThis results in covering more than 92\\% of all the UTXOs per wallet ID.\nAlso, as discussed in section~\\ref{sec:new-protocol}, by using different mapping, one can cover more percentage of Bitcoin addresses.\n\\begin{figure}[b]\n\t\\centering\n\n\t\\includegraphics[width=.9\\linewidth]{fig\/percentage.pdf}\n\t\\caption{Number of transactions per wallet ID. By allowing each address can have up to 2 UTXO, \\mbox{$T^3$}\\xspace can cover approximate $92\\%$ of the UTXO set.} \n\t\\label{fig:utxo-analysis}\n\\end{figure}\n\n\\paragraph{Hardware.}\nWe evaluated the performance of $T^3$\\xspace on a desktop which is equipped with Intel(R) Xeon(R) Silver 4116 CPU @ 2.10GHz, 128GB RAM.\nSince Intel(R) Xeon(R) silver 4116 is not SGX-enabled CPU, we obtain the performance results by running our implementation in the simulation mode. \nHowever, we expect to not have much of a performance difference when executing in the two different modes.\nMore specifically, we have tested the performance of \\mbox{$T^3$}\\xspace\nusing a smaller ORAM tree \nin the hardware mode on a commodity desktop equipped with SGX-enabled Intel Core i7.\nComparing the hardware and simulation mode results (i.e., simulation on the Intel Core i7 CPU),\nwe see no noticeable difference in the running time of both \\textit{\\textit{read-once}}\\xspace and standard ORAM accesses.\n\n\\subsection{Experimental Results}\n\\label{sub:perforamce_analysis}\nWe have implemented \\mbox{$T^3$}\\xspace using multiple threads. \nAs reported in \\cite{multiple-thread-sgx,tramer2018slalom}, as long as the total amount of memory used by all threads does not exceed the EPC limit, \nthe performance gain should be similar to the use of different enclaves. \nIn this work, we implemented all functionalities in one single enclave, and we used multiple threads to concurrently accesses the ORAM trees.\n\\iffalse\n\\begin{asparaenum\n\t\\item \n\t\\textit{The \\textit{managing} enclave\\xspace:}\n\t{\n\t\tAs discussed previously, we used \\texttt{libjson-rpc-cpp}~\\cite{libjson-rpc} to handle the communication between the enclave and the Bitcoin daemon. \n\t\tTo check the integrity of the Bitcoin block, the \\textit{managing} enclave\\xspace verifies the Merkle root and the proof of work of the Bitcoin block using the stored Bitcoin header chain.\n\t\n\t\n\t\tOnce the block is verified, the enclave runs a filter to extract the relevant information of transactions from the block.\n\t\tThe filter will parse the hexadecimal codes representing every transaction and reduces the inputs and outputs to\n\t\t68 byte, each with five fields: the related transaction hash, the amount of bitcoin, the \n\t\tblock height, the public key hash, and the position of the input\/output on the corresponding transactions. \n\t\n\t\tThese data fields are sufficient for a SPV client to form new transactions. \n\t\tOnce the verification algorithm finishes, the \\textit{managing} enclave appends the header of the new block into the Bitcoin header chain and signal the \\textit{writing} enclave to perform update. \n\t\n\t\n\t\n\t\n\t\n\t}\n\t\\item \n\t\\textit{The \\textit{writing} enclave\\xspace:}\n\t{\n\t\n\t\n\t\tThe \\textit{managing} enclave\\xspace redirects all the update operations to the \\textit{writing} enclave\\xspace. \n\t\tTo update each input (i.e., recent spent output) to \\textit{writing} enclave\\xspace, it \n\t\n\t\tfetches the ORAM block containing the related old UTXO and deletes it. \n\t\tNext, the enclave starts adding the new outputs (UTXOs) into the \\textit{original} ORAM tree\\xspace.\n\t\tWhen all of the tree updates are done, the \\textit{managing} enclave will use the \\textit{original} ORAM tree\\xspace as the new \\textit{read-once} ORAM tree\\xspace while keeping a copy for the new \\textit{original} ORAM tree\\xspace.\n\t}\n\t\\item \n\t\\textit{The \\textit{reading} enclave\\xspace:}\n\t{\n\t\n\t\tFor every new SPV client's request, the \\textit{reading} enclave\\xspace reads the block from the \\textit{read-once} ORAM tree\\xspace without performing the eviction routine. \n\t\tAs discussed in the previous section, the eviction routine will be handled by the \\textit{writing} enclave\\xspace.\n\t\n\t\tTo increase the performance of the reads, \n\t we implemented multiple threads on our \\textit{reading} enclave\\xspace and \n\t provided a Stash and a copy of the temporal ORAM tree's position map to each thread.\n\t \n\t\t\n\t}\n\\end{asparaenum}\n\\fi\n\n\n\\paragraph{System parameters}\nWe tested our system with both recursive \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace~and recursive \\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace~using different tree size $N=2^{20}, 2^{21}, 2^{22}, 2^{23}, 2^{24}$. \nWe allow each Bitcoin address to have up to $2$ unspent transaction outputs, \nand we use the single address into single ORAM block mapping approach described in section~\\ref{subsec: Oblivious Storage of the UTXO set} to map addresses into ORAM block. \nFinally, we use claim~\\ref{claim:addressesperoramblock} to determine the size of each ORAM block. \n\n\n\\paragraph{Performance of \\textit{\\textit{read-once}}\\xspace and standard ORAM accesses.} \nIn $T^3$, the \\textit{reading} enclave\\xspace performs \\textit{\\textit{read-once}}\\xspace accesses to handle client's requests in an efficient manner. \nTable~\\ref{table:path-circuit-oram} presents an overall performance of a standard ORAM access as well as the performance of a \\textit{\\textit{read-once}}\\xspace access for both \\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace~and \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace. \nFor this experiment, we took the average running time of 10000 accesses.\n\n\nAs shown in the results,\nORAM constructions with smaller block sizes provides a better performance in both schemes.\nThe reason is that oblivious operations like oblivious comparisons and $\\mathsf{cmov}$-based stash scan are more efficient because of a smaller size stash. \nMoreover, \\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace gives a better performance compared to \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace, as it can operate on a smaller block compared to \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace, and this requires much smaller stash size\nallowing much faster oblivious execution.\n\\begin{table}[b]\n\\centering\n\\resizebox{.95\\columnwidth}{!}{%\n\\begin{tabular}{c|c|c|}\n\\cline{2-3}\n & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\mbox{$T^3$}\\xspace(\\textbf{\\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace})}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\mbox{$T^3$}\\xspace(\\textbf{\\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace})}}\\\\\\cline{1-1}\n \\multicolumn{1}{|c|}{\\textbf{Number of threads}} & \t\t\t & \\multicolumn{1}{c|}{} \t\t\\\\ \\hline\n \\multicolumn{1}{|c|}{$1$}\t \t\t\t\t\t\t\t\t& {2.43} ms & \\multicolumn{1}{c|}{0.64 ms} \\\\ \\hline\n \\multicolumn{1}{|c|}{$2$} \t\t\t\t\t\t\t\t\t& {1.40} ms & \\multicolumn{1}{c|}{0.58 ms} \t\\\\ \\hline\n \\multicolumn{1}{|c|}{$3$} \t\t\t\t\t\t\t\t\t& {0.90} ms & \\multicolumn{1}{c|}{0.43 ms} \t\\\\ \\hline \n \\multicolumn{1}{|c|}{$4$} \t\t\t\t\t\t\t\t\t& {0.73} ms & \\multicolumn{1}{c|}{0.35 ms} \t\\\\ \\hline\n\n\\end{tabular}\n}\n\\vspace{10pt}\n\\captionof{table}{Performance gain of multiple-thread \\textit{\\textit{read-once}}\\xspace access on Path\/\\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace with $N=2^{24}$ block size = $544$ bytes.}\n\\label{table:path-circuit-oram-multiple-threading}\n\\end{table}\n\n\n\\paragraph{Parallelization.} Since there is no race condition in the \\textit{\\textit{read-once}}\\xspace accesses, the design of $T^3$\\xspace allows different threads to concurrently perform \\textit{\\textit{read-once}}\\xspace accesses on the \\textit{read-once} ORAM tree\\xspace. \nCompared to other oblivious system like \\textsc{Bite}~\\cite{SasyGF18-zero-trace}, $T^3$\\xspace is able to handle bursty client read requests concurrently while the eviction requests are distributed sequentially during the \\textit{block creation interval}\\xspace. \nTo measure this performance gain, we used multiple threads to access the \\textit{\\textit{read-once}}\\xspace enclave and perform \\textit{\\textit{read-once}}\\xspace access simultaneously on a tree of size $N=2^{24}$ and ORAM block of size $544$ bytes. \nTable~\\ref{table:path-circuit-oram-multiple-threading} shows the performance of \\mbox{$T^3$}\\xspace implemented using multiple threads for both \\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace~and \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace.\n\n\n\\paragraph{Comparison to current SPV solutions. }\nWe give a comparison in term of performance and communication overhead over several number of requests to the existing SPV client's solution and to BITE~\\cite{bite-spv-sgx}\nOblivious database.\n\\begin{asparaenum}\n\t\\item \\textit{Performance}:\n\t\\Cref{fig:performance} gives us an overview of the performance of \\mbox{$T^3$}\\xspace compared to the performance of the current existing SPV with Bloom filter solution and the performance of Bite Oblivious database. \n\tIn particular, it shows the response latency from the client's perspective. \n\n\n\tIn this comparison, a request for the SPV solution with Bloom filter solution means the time the server takes to scan one Bitcoin block, \n\tand a request for \\mbox{$T^3$}\\xspace and BITE means the time it takes to perform an ORAM access on the ORAM tree. \n\t\tFor the current SPV clients with Bloom filter, we set the false positive rate of the Bloom filter to $1.0$\\% and $5.0$\\% respectively. \n\tFor \\mbox{$T^3$}\\xspace, we used $N=2^{24}$ and block of size 544 bytes for both \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace with $Z=4$ and \\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace with $Z=2$.\n\tFor BITE database, based on our understand of their construction, we re-implemented BITE using non-recursive construction of \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace, and we used the same ORAM block of size $32$kB which leads to the number of block is $N=2^{17}$. \n\tAlso, we also provide an additional construction of BITE which is implemented using recursive \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace and suggested parameters for \\mbox{$T^3$}\\xspace where the tree is of size $2^{24}$ and block of size $544$B. \\Cref{fig:performance} gives us the overall performance of three existing solutions. \n\t\n\tThe performance of \\mbox{$T^3$}\\xspace outperforms the SPV with Bloom filter solution. \n\tThe reason is that in \\mbox{$T^3$}\\xspace, the system relies on the TEE to handle the integrity checking of the Bitcoin block before updating the ORAM tree while in the current SPV solution, the full client needs to scan the Bloom filter every time and detect the relevant transactions and recompute the Merkle path for each of those transactions. \n\n\t\n\tAlso, \\mbox{$T^3$}\\xspace performs much better than BITE oblivious database as the BITE system does not consider the use of recursive ORAM construction. \n\n\n\tAnother reason is that the size of the ORAM block used in Bite is large; hence, the cost of oblivious operation like $\\mathsf{cmov}$-based stash scan becomes more expensive.\n\tThus, we envision and realize an improved construction of BITE using recursive construction of \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace to demonstrate the practical impact of using recursive ORAM construction on TEE with restricted memories. \n\n\n\n\t\\begin{figure}[b]\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\linewidth]{fig\/performance-spv-oram.pdf}\n\n\t\t\\caption{Performance of $T^3$ using \\textsc{Path}\/\\textsc{Cicruit} ORAM with block of size $544$B, the current SPV with Bloom filter, Original BITE oblivious database block of size $32$kB, and improved BITE with block of size $544$B. For the SPV client with Bloom filter, we used the false positive rate of $1\\%$ and $5\\%$.}\n\t\t\\label{fig:performance}\n\t\\end{figure}\n\n\n\t\\item \\textit{Communication Overhead}: \n\tIn term of communication between client and server, \\mbox{$T^3$}\\xspace offers much lower communication overhead compared to the existing solution for SPV clients. \n\t\\mbox{$T^3$}\\xspace does not need to provide the SPV clients with the Merkle proofs to its relevant transactions because all those proofs are validated by the Intel SGX before being added the ORAM tree. \n\tThus, one can reduce both the amount of work that the full node needs to perform and the amount of data that it needs to send to the SPV clients. \n\tMoreover, \\mbox{$T^3$}\\xspace prunes all other information of transactions to extract only relevant data needed for client to determine balance and form new transactions, \n\n\n\twhile in the current SPV solution, due to the false positive rate used in the Bloom filter, the full client may send additional irrelevant information to the SPV client. \n\t~\\Cref{fig:communication} shows an overview of the communication cost of \\mbox{$T^3$}\\xspace compared to the current solution. \n\tTo give an estimation of the communication cost of the current SPV solution, we assumed that each request requires a separate Merkle proof. \n\tHence, for each request, the size of the proof is at least: $\\log_2(\\textsf{NoTXs})\\cdot 32$ bytes where $\\mathsf{NoTXs}$ is the number of transactions in one block.\n\tMoreover, the size of the transaction data is approximately $\\mathsf{fpr} \\cdot \\textsf{BlockSize}$~bytes where the $\\mathsf{fpr}$ is the false positive rate and the $\\mathsf{BlockSize}$ is the size of the Bitcoin block.\n\tTo compute the overhead cost we used block 551731, as an example, which has block size of 1149 KB and contains 3017 transactions. \n\tHowever, in practice, we would expect the Bitcoin blocks to have different sizes; resulting, the communication cost to be different across blocks.\n\tTherefore, the results in \\cref{fig:communication} is only an estimation on the communication overhead using the current SPV solution. \n\tWe omit the comparison to the communication overhead of BITE because both \\mbox{$T^3$}\\xspace and BITE return a fixed amount a data to the SPV client which is the output itself. \n\t\\begin{figure}[t]\n\t\t\\centering\n\t\n\t\t\\includegraphics[width=\\linewidth]{fig\/communication-spv-oram.pdf}\n\n\t\t\\caption{Communication cost of $T^3$ and the current SPV solution. Since both systems return the information of unspent outputs to the client, the communication overhead of BITE will be equal to the communication overhead of \\mbox{$T^3$}\\xspace.} \n\t\t\\label{fig:communication}\n\t\\end{figure}\n\\end{asparaenum}\n\n\n\\paragraph{Storage Overhead}\nAs noted in the previous section, using ORAM incurs a constant size blow up of the storage of the UTXOs (e.g., $\\approx 3-4\\times$ for \\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace, $6-8\\times$ for \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace). \nIn particular, for \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace with $Z=4$, the storage cost of ORAM trees is about $\\approx 51GB$, \nand for \\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace with $Z=2$, the storage cost of two ORAM tree is around $\\approx 26GB$. \nFor the EPC memory usage, we need to consider the size of the position map and the stash used by each enclave. \nIn this work, since we use recursive ORAM constructions for both schemes, \nthe size of the position map can be as small as possible at the cost of storing more recursive ORAM trees in the untrusted memory region. \nPrecisely, in the prototype of our implementation, each thread uses $8$KB for the position map and a stash of size $2\\cdot \\log(N)\\cdot Z \\cdot \\mathsf{BlockSize}$ bytes (e.g., for a tree of size $N=2^{24}$, we use $\\approx0.62$MB bytes for \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace, and $\\approx 0.31$ MB for \\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace). \nThus, as long as the total memory usage by all threads\/enclaves does not exceed the EPC limit (e.g., $96$MB), the performance of the system will not suffer from the expensive swapping operations discussed in section~\\ref{sub:intel_sgx}. Moreover, for integrity protection, \\mbox{$T^3$}\\xspace only requires the server to store the Bitcoin header chain which is approximately $44$MB instead of storing a complete Bitcoin blockchain. Thus, in the future work, if \\mbox{$T^3$}\\xspace can handle the communication with the Bitcoin network without the reliance on the existing Bitcoin client, \\mbox{$T^3$}\\xspace reduces the need of storing the 230 GB of Bitcoin blockchain.\n\n\\subsection{Comparison with Other Oblivious Systems}\nHere we provide a comparison between \\mbox{$T^3$}\\xspace and other generic oblivious systems. We compare our work with $\\textsc{Bite}$~\\cite{bite-spv-sgx} Oblivious Database \nthat also uses ORAM and TEE to provide a generic PIR system for Bitcoin client, $\\textsc{ConcurORAM}~\\cite{concurORAM-chakraborti}$ that provides concurrency access to ORAM clients, $\\textsc{Obliviate}~\\cite{ndss-AhmadKSL18}$ that prevents leakage from file system accesses, and $\\textsc{ZeroTrace}$ which proposes an efficient generic oblivious memory access primitives.\nIn particular, \\cref{table:comparision} compares those systems based on the capabilities of supporting concurrency access, enabling recursive construction, and preventing side-channel leakage.\n\n\\begin{table}[b]\n\\centering\n\\begin{minipage}{\\columnwidth}\n\\resizebox{.98\\columnwidth}{!}{%\n \\begin{tabular}{lccc}\n \\toprule\n & \\multicolumn{3}{c}{Capabilities}\\\\\n \\cmidrule(lr){2-4}\n System \t\t\t & Concurrency & Recursive Construction & Side-channel Protection\\\\\n \\midrule\n \\textsc{ConcurORAM}~\\cite{concurORAM-chakraborti}\t\t& \\cmark & \\xmark & -~\\footnote{\\textsc{ConcurORAM} does not aim to provide side-channel protection for TEE. Hence, we omit this comparison.} \\\\\n \\textsc{Obliviate}~\\cite{ndss-AhmadKSL18} \t\t\t& \\xmark & \\xmark & \\cmark \\\\\n \\textsc{Zerotrace}~\\cite{SasyGF18-zero-trace} \t\t& \\xmark & \\cmark & \\cmark \\\\\n \\textsc{Bite} Oblivious Database~\\cite{bite-spv-sgx} \t& \\xmark & \\xmark & \\cmark \\\\\n \\mbox{$T^3$}\\xspace \t \t\t\t\t\t\t\t\t\t\t& \\cmark & \\cmark & \\cmark \\\\\n \\bottomrule\n \\end{tabular}\n}\n\\end{minipage}\n\\label{table:comparision}\n\\caption{Comparison between \\mbox{$T^3$}\\xspace and other oblivious systems.}\n\\end{table}\n\nFor generic trusted hardware-based systems like \\textsc{Bite} oblivious database and \\textsc{Obliviate}, while providing protection against side-channel leakage, those systems do not consider the use of recursive ORAM construction to reduce the EPC memory usage. \nHence, the performance of their systems will degrade once the database becomes too large. \nOther works that harnesses the use of recursive ORAM construction are \\textsc{Zerotrace}; however, concurrency is not supported in the current version of \\textsc{Zerotrace}. Thus, without concurrency support, such systems will not scale well to handle Bitcoin SPV clients.\n\\textsc{ConcurORAM} is a recent ORAM construction that offers concurrency accesses from the clients; however, due to more optimized eviction strategy and complex synchronization schedule, the recursive construction of \\textsc{ConcurORAM} introduces implementation challenges. \nNevertheless, we believe that it can be an interesting future work to use \\textsc{ConcurORAM} in the design of \\mbox{$T^3$}\\xspace. \n\n\n\\section{Preliminaries and Threat Model}\n\\label{sec:preliminaries}\n\\subsection{Trusted Execution Environment}\n\\label{sub:intel_sgx}\nThe design of $T^3$\\xspace relies on a trusted execution environment (TEE) to prove the correctness of the computations. \nIn particular, TEE is a trusted hardware that provides both confidentiality and integrity of computations as well as offer an authentication mechanism, known as \\textit{attestation}, for the client to verify computation correctness.\nIn this work, we chose Intel SGX~\\cite{sgx-explained} to be the building block of our system.\nHowever, with minor modifications, the design of our system can be extended to any TEE with \\textit{attestation} capabilities such as Keystone-enclave~\\cite{keystone-project} and Sanctum~\\cite{constant-sanctum-usenix} as other trusted execution environments might not have the same strengths\/weaknesses as Intel SGX.\n\nIntel SGX is a set of hardware instructions introduced with the 6th Generation Intel Core processors. \nWe use Intel SGX as a TEE for the execution of an ORAM controller on the untrusted server. \nThe relevant elements of SGX are as follows.\n\t\\textbf{Enclave} is the trusted execution unit that is located in a dedicated portion of the physical RAM called the enclave page cache (EPC). The SGX processor makes sure that all other software components on the system cannot access the enclave memory. \n\tIntel SGX supports both {\\bf local and remote attestation} mechanisms to allow remote parties or local enclaves to authenticate and verify if the program is correctly executed within an SGX context.\n\tMore importantly, attestation protocols provide the authentication required\n\tfor a key exchange protocol \\cite{sgx-explained}, i.e.,\n\tafter a successful attestation, the concerned parties can agree on a shared session key using Diffie-Hellman Key Exchange~\\cite{DH-keyexchange} and create a secure channel.\n\n\n\n\\noindent\\textbf{Limitations.}\nIntel SGX comes with various limitations which have been uncovered by the\nacademic community over the past few years.\nIn particular, some of the limitations are:\n\\begin{compactitem}\n\t\\item \\textbf{Side Channel Attacks:} While Intel SGX provides security guarantees against direct memory attacks, it does not provide systematic protection mechanisms against side channel attacks\n\tsuch as page table-based~\\cite{Xu15ControlledChannel, hid-sgx-sidechannel-usenix17}, cache-based~\\cite{cache-based-attack}, and branch-prediction-based~\\cite{shadow-branch-lee-usenix17}.\n\tThrough page table and cache attacks, a privileged attacker can\n\tobserve cache-line-granular (i.e., 64B) memory access patterns from the enclave program. On the other hand, the branch-prediction attack\n\tcan potentially leak all the control-flow taken by the enclave program.\n\n\t\\item \\textbf{Enclave Page Cache Limit:} The size of the Enclave Page Cache (EPC) is limited to around $96$MB~\\cite{Arnautov-epc}. Although Intel SGX alleviates this limitation by supporting page-swapping between trusted memory region and untrusted memory region, this operation is expensive due to encryption and integrity verification \\cite{sgx-explained,Arnautov-epc}. \n\n\t\\item \\textbf{System Calls:} Intel SGX programs are restricted to ring-3 privileges and therefore rely on the untrusted OS for ring-0\n\toperations such as file and network I\/O. \n\tThere are various previous works which try to solve this problem using library OSes~\\cite{lib-oses} \n and\/or other techniques~\\cite{thang-hoang:posup-popets}. \n\\end{compactitem}\n\n\n\\noindent\n\\textbf{Oblivious Operations inside the Enclave.}\n\tSeveral techniques~\\cite{racoon,thang-hoang:posup-popets,SasyGF18-zero-trace,oblivious-olga-ohrimenko} have been introduced to mitigate side-channel attacks on the SGX. \n\n\tIn this work, we built our system based on the implementations of both $\\mathsf{Zerotrace}$~\\cite{SasyGF18-zero-trace} and $\\mathsf{Obliviate}$~\\cite{ndss-AhmadKSL18}.\n\t%\n\tTherefore, our system inherited standard secure operations from both of these libraries. \n %\n In particular, their implementations use an oblivious access wrapper\n by using the x86 instruction $\\mathsf{cmov}$\\xspace as introduced by Raccoon~\\cite{racoon}.\n %\n Using $\\mathsf{cmov}$\\xspace, the wrapper accesses every single byte of a memory object\n while reading or writing only the required bytes in memory.\n %\n From the perspective of an attacker (which can only observe\n access-patterns), this is the same as reading or modifying\n every byte in memory.\n %\n\tWe refer readers to \\cite{SasyGF18-zero-trace,ndss-AhmadKSL18, racoon} \n\tfor detailed description of these oblivious operations.\n\n\\subsection{Oblivious Random Access Memory}\n\\label{sub:oblivious_ram}\nOblivious Random Access Memory (ORAM) was first introduced by Goldreich et al~\\cite{Goldreich:1987} for software protection against piracy. The core of ORAM is to hide the access patterns resulted by reading and writing accesses on encrypted data. The security of ORAM can be described as follows.\n\n\\begin{definition}~\\cite{Stefanov:2013} \n\tLet\n\t$\\stackrel{\\rightarrow}{y}=(\\mathsf{op_i,bid_i,data_i})_{i\\in [n]}$\n\tdenote a sequence of accesses \n\twhere $\\mathsf{op_i}\\in \\{read,write\\}$, \n\t$\\mathsf{bid_i}$ is the identifier, \n\tand $\\mathsf{data_i}$ denotes the data being written. \n\tFor an ORAM scheme $\\Sigma$, let $\\mathsf{Access}_{\\Sigma}(\\stackrel{\\rightarrow}{y})$ denote a sequence of physical accesses pattern on encrypted data produced by $\\stackrel{\\rightarrow}{y}$.\n\tWe say:\n\t\\begin{inparaenum}[(a)]\n\t \t\\item The scheme $\\Sigma$ is secure if for any two sequences of accesses $\\stackrel{\\rightarrow}{x}$ and $\\stackrel{\\rightarrow}{y}$ of the same length, $\\PAccess{\\ORequest{x}}$ and $\\PAccess{\\ORequest{y}}$ are computationally indistinguishable.\n\t \t\\item The scheme $\\Sigma$ is correct if it returns on input $\\ORequest{y}$ data that is consistent with $\\ORequest{y}$ with probability $\\geq 1 - \\mathsf{negl}(\\lvert\\ORequest{y}\\rvert)$ i.e {negligible in $\\lvert\\ORequest{y}\\rvert$}\n \t\\end{inparaenum}\n\\end{definition}\n\n\\noindent\n\\textbf{Tree-based ORAM schemes. } One strategy of designing an ORAM scheme is to follow the tree paradigm proposed by Shi et al.~\\cite{Elaine-rORAM} and Stefanov et al.~\\cite{Stefanov:2013}. \nIn tree based ORAM, the client encrypts their database into $N$ different encrypted data blocks and obliviously stores those data blocks in a binary tree of height $\\lceil \\log_2(N) \\rceil$. \nEach node in the tree is called a \\textit{bucket}, and each \\textit{bucket} can contain up to $Z$ blocks. \nThe client also maintains a \\textit{position map}, to indicate which path a data block resides on. \nFinally, the client needs to have a \\textit{stash} to store a path retrieved from the server.\n\nWe follow the same generalization of a tree-based ORAM access described in~\\cite{thang-hoang:posup-popets}. \nEach access in both ORAM schemes requires two operations: a $\\mathsf{ReadPath}$ operation and an $\\mathsf{Evict}$ operation. \nIntuitively, $\\mathsf{ReadPath}$ takes as input the ORAM block identifier, $\\mathsf{bid}$, accesses the position map, and retrieves the path that block $\\mathsf{bid}$ resides onto the stash, $S$. \nAfter performing ORAM access (i.e. $\\mathsf{read\/write}$) on the identified block, the block is assigned to a different path and pushed back to the tree via the $\\mathsf{Evict}$ operation.\nIn general, the $\\mathsf{Evict}$ operation takes a stash and the assigned path as input, writes back blocks from stash to the assigned path, and update the position map. \nFigure~\\ref{fig:oram-access} gives an overview of how tree-based ORAM access works.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{minipage}{\\linewidth}\n\t\\begin{algorithm}[H]\n\t \t\\caption{$ORAM.\\mathsf{Access}(\\mathsf{op,bid,data^*})$}\n\t \t\\begin{algorithmic}[1]\n\t\t\n\t\t\t\\State $S \\leftarrow \\mathsf{ReadPath}(\\mathsf{bid})$ \n\t\t\t{\\color{blue}{\/\/scan the whole stash}}\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\t\\State $\\mathsf{data}\\leftarrow$ block $\\mathsf{bid}$ from $S$ \n\t\t\t\\If{$\\mathsf{op} = write$}\n\t\t\t\\State $S$ $\\leftarrow$ $(S$$ - \\mathsf{\\{(bid,data)\\})\\cup \\{(bid,data^*)\\}}$\n\t\t\t\\EndIf\n\t\t\t\\State $p' \\stackrel{\\$}{\\leftarrow} \\{0, \\dots, N-1\\}$ {\\color{blue}{\/\/the random eviction path is selected}}\n\t\t\t\\State $S\\leftarrow \\mathsf{Evict}(S,p')$ \n\t\t\n\t\t\n\t\t\n\t\t\n \t\n\t\t\n\t\t\n\t\t\t\\State \\Return $\\mathsf{data}$\n\t\t\\end{algorithmic}\t\n\t\\end{algorithm}\n\t\\end{minipage}\n\t\\caption{a standard tree-based ORAM \\textit{read\/write} access}\n\t\\label{fig:oram-access}\n\\end{figure}\n\\noindent\n\\textbf{\\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace\/\\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace scheme.}\nIn this work, we consider two popular tree-based constructions of ORAM which are $\\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace$~\\cite{Stefanov:2013} and $\\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace$~\\cite{wang-circuit-oram-2015}. \nWhile $\\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace$ offers simple $\\mathsf{ReadPath}$ and $\\mathsf{Evict}$ operations, $\\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace$ offers a smaller circuit complexity for the $\\mathsf{Evict}$ procedure. \nThus, $\\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace$ is more efficient when implemented with Intel SGX. \nAs noted in \\cite{SasyGF18-zero-trace,thang-hoang:posup-popets,wang-circuit-oram-2015}, $\\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace$ can operate with $Z=2$ compared to $Z=4$ as in $\\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace$; therefore, the server storage overhead is significantly reduced. Moreover, the size of \\textit{stash} in $\\ensuremath{\\textsc{Circuit\\text{-}ORAM}}\\xspace$ is smaller compared to the size of \\textit{stash} in $\\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace$; this allows a more efficient performance when scanning the stash as one needs to scan the whole path and stash to avoid side-channel leakage.\n\\noindent\n\n\\noindent\n\\textbf{Recursive ORAM.} \nIn a non-recursive tree-based ORAM setting, the client has to store a position map of the size $O(N)$ bits. \nThis approach, however, is not suitable for a resource-constrained client. \nStefanov et. al~\\cite{Stefanov:2013} presented a technique that reduces the size of the position map to $O(1)$. \nThe main idea of those constructions is to store a position map as another ORAM tree in the server side, and the client only keeps the position map of the new ORAM.\nThe client keeps compressing the position map into another ORAM tree until the size of the position map is small enough to be saved on the client's storage. \nOne main drawback of those constructions is the increased cost in the communication between a client and the server. \nFortunately, in our setting, this cost can be safely ignored because the communication between client and server becomes the I\/O access between TEE and the random access memory.\nThus, it is more reasonable to use recursive constructions because it reduces the memory stored in the trusted region (e.g., Processor Reserved Memory).\n\n\\subsection{Blockchain}\n\\label{sub:btc}\n\nThe Bitcoin blockchain is a distributed data structure maintained by a group of nodes. \nIn this work, to simplify the structure of the Bitcoin blockchain, we denote the network as a single party that maintains a growing database of Bitcoin blocks.\nOn average of $10$ minutes, the network outputs a Bitcoin block which is a combination of Bitcoin transactions and a block header.\nEach block header contains relevant information about the Bitcoin block such as Merkle root, nonce, network difficulty.\nThe Merkle root can be used to verify the membership of Bitcoin transactions, \nand the nonce and difficulty are used to check the proof of work.\nEach Bitcoin transaction contains a set of inputs and outputs where transaction inputs are unused outputs of previous transactions.\n\t\n\\begin{asparaitem\n\t\\item\\textbf{Unspent Transaction Output Database.}\t\n\t\tIn the Bitcoin network, the balance of a Bitcoin address is determined by values of those outputs that have not been used in other transactions.\n\t\tThese outputs are called Unspent Transaction Outputs (UTXO).\n\t\tMoreover, in the implementation of common Bitcoin nodes such as Bitcoin core \\cite{bitcoin-core}, \n\t\n\t\tBitcoin nodes maintain a separate database that keeps track of all unspent transaction outputs and other metadata of the Bitcoin blockchain. \n\t\tThis database is known as the UTXO set.\n\t\n\t\n\t\n\t\tIntuitively, a client with the knowledge of the secret key and the commitment value can query the UTXO set directly to obtain essential information such as transaction hash, position, and value to form new valid transactions.\n\t\n\t\tTherefore, in this work, we realize that if a full node can securely update and maintain the integrity of the UTXO set via while provides SPV clients with oblivious accesses to the UTXO set, the privacy of the SPV client is preserved.\n\n\t\\item \\textbf{Bitcoin transaction types.}\n\tIn the Bitcoin blockchain, transactions are classified based on the structure of the input and output scripts. \n\tIn particular, there are five types of standard script templates which are \\textit{Pay-to-Pubkey} (P2PK), \\textit{Pay-to-PubkeyHash} (P2PKH), \\textit{Pay-to-ScriptHash} (P2SH), \\textit{Multisig}, and \\textit{Nulldata}. \n\tIntuitively, scripting in Bitcoin provides a way to prove the ownership of the coins. \n\tIn particular, a challenge script (\\textit{scriptPubkey}) is included as a part of the transaction output to specify the condition for its redemption, and a response script (\\textit{scriptSig}) is part of the transaction input to reveal the condition needed to redeem the bitcoins from other output. \n\n\tIn this work, we only consider two types of transaction: \\textit{Pay-to-PubkeyHash} (P2PKH) transaction and $\\textit{Pay-to-ScriptHash}$ (P2SH) transaction.\n\tAccording to \\cite{analysis-of-utxo,mohsen-r3c3}, these two types of transaction make up of $97\\text{-}99\\%$ of the UTXO set. \n\tAlso, one can assume that the \\textit{Pay-to-Pubkey-Hash} transaction is one variant of the \\textit{Pay-to-Script-Hash} transaction \n\tbecause both transaction types require the spender's knowledge of the preimage of the hash digest before being able to spend those outputs. \n\tFor simplicity, from this point on, we assume that the only information needed to obtain the unspent output is the public key hash, $pkh$. \n\tMoreover, all other transaction types such as \\textit{Multisig} and P2PK can be easily supported in the future.\n\n\t\n\t\\item \\textbf{Block creation interval. } The block creation time in Bitcoin is the time that the network takes to generate a new block, and block creation time is specified to be 10 minutes on average by the network. \n\tWe call the waiting period between the most recent block and a new block, \\textit{block creation interval}. \n\tIn this work, we discretize time as \\textit{block creation intervals}. \n\\end{asparaitem}\n\n\n\n\\subsection{Threat Model}\n\\label{sub:Threat_Model}\nWe assume that SPV \\textit{clients} are honest and rational which means that before during the \\textit{block creation interval},\na SPV \\textit{client} should not request the server for transaction outputs of a same public key hash more than once. \nThe underlying remote attestation service provided by Intel is secure and trusted. \nThe local attestation between enclaves is secure. \nThe server and its programs are assumed to be untrusted except for programs running within an enclave. \n\nWe assume that the adversary who controls the operating system can read\/inject\/modify encrypted messages sent by enclaves. \nThe adversary also can observe memory access patterns of both trusted and untrusted memory. Also, the computation power of the adversary is assumed to be limited. \nIn particular, during \\textit{block creation interval}, the adversary should not have enough computation power to forge a new Bitcoin block that satisfies the current Bitcoin network difficulty. As the time of writing, the network difficulty~\\cite{bitcoin-difficulty} is around $6\\times 10^9$; \ntherefore, the expected number of hashes to mine a Bitcoin block is roughly $2^{72}$. \n\nThe server's attacks on availability are out of scope.\nMore specifically, denial of service (DoS) attacks by system admin and untrusted operating system are out of the scope.\nOtherwise, such adversary can prevent the enclaves from receiving new bitcoin block by shutting down the communication channel between the enclave and the Bitcoin network as the enclave has to rely on the untrusted OS to perform system calls such as file and network I\/O. \n\n\n\n\n\\section{Related Work}\n\\label{sec:related-work}\n\n\\paragraph{General SGX Systems.}\nHaven~\\cite{haven} is a pioneering work on SGX computing\nenabling native application SGX porting on windows.\nGraphene~\\cite{graphene} provides a linux-based LibOS for\nSGX programs.\nRyoan~\\cite{ryoan} retrofits Native Client to provide sandboxing\nmechanisms for Intel SGX.\nEleos~\\cite{eleos} provides a user-space extension of enclave\nmemory using custom encryption.\n\\mbox{$T^3$}\\xspace uses some concepts from Eleos especially in the way we store\nthe ORAM tree using custom encryption outside the SGX enclave.\n\n\\paragraph{SGX Side-channels.}\nThere are three main memory-based side-channel vulnerabilities disclosed\nwithin Intel SGX, namely, page table-based attacks~\\cite{Xu15ControlledChannel},\ncache-based attacks~\\cite{cache-based-attack}, and branch-prediction\nattacks~\\cite{hid-sgx-sidechannel-usenix17}.\nFurthermore, since SGX relies on the untrusted OS for system-call\nhandling, it is also vulnerable to IAGO attacks~\\cite{iago-attack}.\nLeaky Cauldron~\\cite{leaky-cauldron} presents an overview of the possible attack vectors\nagainst SGX programs.\n\\mbox{$T^3$}\\xspace is secure against all disclosed memory-based side-channels since it\nuses oblivious RAM (ORAM) to protect the access-patterns.\nFurthermore, \\mbox{$T^3$}\\xspace uses oblivious memory primitives to secure the runtime\nORAM operations as well as its library.\n\n\\paragraph{Oblivious Systems.}\nRaccoon~\\cite{racoon} provided a technique to protect a small part of\na user program against all digital side-channels.\n\\textsc{Obliviate}~\\cite{ndss-AhmadKSL18} and \\textsc{ZeroTrace}~\\cite{SasyGF18-zero-trace}\nused ORAM-based operations to protect files and data arrays respectively\ninside Intel SGX. \nThang Hoang et al.~\\cite{thang-hoang:posup-popets} proposed a combination of TEE and ORAM to design oblivious search and update platform for large dataset.\nEskandarian et a.~\\cite{oblidb} leveraged Intel SGX and Path ORAM to propose oblivious SQL database management system.\n\nRecently, Chakraborti et al. proposed a new parallel ORAM scheme called ConcurORAM~\\cite{concurORAM-chakraborti}. \nSimilar to the \\mbox{$T^3$}\\xspace design, ConcurORAM also uses two-tree structure to propose a non-blocking eviction procedure, and the system periodically synchronizes two trees to maintain the privacy of the user's access pattern. \nIn ConcurORAM, the scheme requires the client to download the query log and the result log to learn about ongoing queries before requesting ORAM accesses.\nHence, if we combine ConcurORAM along with Intel SGX to design this system, the use of query and result logs introduces additional storage overhead to the limited storage capacity of the Intel SGX.\nMore importantly, the author also noted that ConcurORAM cannot be trivially extended to a recursive ORAM construction because of concurrent data structure accesses. \nHowever, if ConcurORAM can be implemented into a recursive ORAM construction, we believe that ConcurORAM can be an interesting alternate solution for the ORAM scheme used in the design of \\mbox{$T^3$}\\xspace. \n\nAnother interesting parallel ORAM construction is TaoStore~\\cite{taostore-sahin}. \nTaoStore assumes a trusted proxy that handles concurrent client's requests, and the proxy runs a scheduler to make sure that there are no conflicting queries while preventing no information leakage. \nHowever, similar to ConcurORAM, the implementation of TaoStore is limited to the non-recursive construction of Path ORAM which is not suitable when combining with TEE with limited trusted memory capacity. \nThis work aims to design a simpler design that is suitable for any flavor of tree-based ORAM schemes. \n\n\n\\paragraph{TEE for cryptocurrencies.} The research community has investigated different ways of combining TEE with blockchain to both improve privacy and scalability of blockchains. \nObscuro~\\cite{obscuro-muoi-tran} is a Bitcoin transaction mixer implemented in Intel SGX that addresses the linkability issue of Bitcoin transactions.\nTeechan~\\cite{teechan} is an off-chain payment micropayment channel that harnesses TEE to increase transaction throughput of Bitcoin. Bentov et al. proposed a new design that uses Intel SGX to build a real-time cryptocurrency exchange. \nAnother example is the Towncrier system~\\cite{towncrier-Zhang} that uses TEE for securely transferring data to smart contract. \nAnother prominent example is Ekiden~\\cite{ekiden-Cheng} which proposed off-chain smart contract execution using TEE. Finally, ZLite~\\cite{ZliTE} system is another example which used ORAM and TEE to provide SPV clients with oblivious access. However, similar to BITE, ZLite employed non-recursive \\ensuremath{\\textsc{Path\\text{-}ORAM}}\\xspace as it is, and thus, the scalability and efficiency of the system is inherently limited due to the non-concurrent accesses.\n\n\n\nOsuntokun et al.~\\cite{osuntokun-client-filter} recently present a new proposal for Bitcoin SPV clients. \nThis proposal is the building block for systems like Neutrino.\nIn particular, each block will have its own Bloom filter. \nThe SPV client first fetches the filter from the full client and decides to download the block from another client if transactions of interested are in the block. \nThis approach, however, introduces an additional communication overhead to the client. \nIn particular, a client with lots of transactions scattered among different blocks needs to download lots of full blocks, and performing verification of block can be expensive for the resource-constrained client. \nThis approach does not necessarily provide more privacy for the SPV client as the full client still learn the block that the addresses belong to.\n\n\n\n\n\\section{System Analysis}\n\\label{sec:security-analysis}\n\n\\subsection{Security Claims}\n\\label{sub:privacy}\nIn order to prove the security properties of \\mbox{$T^3$}\\xspace's design, we put forth six\nclaims, each of which represents the security of a major component of \\mbox{$T^3$}\\xspace in term of privacy goal.\n\n\\paragraph{Claim 1. The managing enclave does not leak user-related information to an attacker.}\nThe managing enclave is responsible for three tasks ---\n(a) converting wallet IDs to UTXOs, (b) creating and managing\nthreads which will perform read operations on the \\textit{read-once} ORAM tree\\xspace, \nand (c) handle the updates to be performed on the \\textit{original} ORAM tree\\xspace.\n\nFirstly, the conversion of wallet IDs to their respective UTXOs is\nprivate since the channel between clients and the \\textit{managing} enclave\\xspace is secured by the shared key during the remote attestation process.\nMore importantly, when receiving addresses from a client, the \\textit{managing} enclave\\xspace uses blockmapping function (described in~\\ref{subsubsec: btcintoORAM}) to map each address to a fixed number of ORAM blocks. \nThis does not reveal information about the number of outputs belonging\nto an address.\nSecondly, each read thread performs the same operations irrespective\nof the wallet ID provided to it, i.e., each thread simply retrieves\nan ORAM block using ORAM accesses implemented with $\\mathsf{cmov}$-based oblivious executions. \nLastly, the only thing revealed by the update process of \\mbox{$T^3$}\\xspace is the\nnumber of blocks updated into the Write Tree. However, this is public\ninformation and \\mbox{$T^3$}\\xspace does not try to hide it.\nEach update is performed using an ORAM access which ensures that the attacker is unaware of the final position of each block.\n\n\\paragraph{Claim 2. The optimized read operations on \\textit{read-once} ORAM tree\\xspace do not leak information.}\nAs explained in section~\\ref{subsub:read-proc}, the \\textit{read-once} ORAM tree\\xspace is accessed using an optimized read\noperation which chooses not to shuffle and write-back the retrieved path to the \\textit{read-once} ORAM tree\\xspace.\nHowever, this is secure since each path corresponding to a UTXO can\nonly be accessed once during a read interval and will be shuffled\nbefore the next interval.\n\n\n\\paragraph{Claim 3. The write operations performed on the \\textit{original} ORAM tree\\xspace\ndo not leak information.}\nThere are two specific operations performed on the \\textit{original} ORAM tree\\xspace --- (a) the UTXOs\nare updated based on the updated bitcoin block, and\n(b) the previously accessed ORAM blocks are shuffled.\nHowever, all of these updating accesses are standard ORAM operations implemented in a side-channel-resistant manners as previously done by~\\cite{SasyGF18-zero-trace,ndss-AhmadKSL18}. Therefore, all write operations reveal no information about a user's UTXO.\n\n\\paragraph{Claim 4. The data fetched from the untrusted world to the TEE is correct.}\nThere are two major sources of data transferred from the untrusted to\nthe trusted world --- (a) the updated block fetched from the bitcoin\ndaemon after a fixed interval and (b) the ORAM tree blocks which are\nfetched from the untrusted world into the TEE.\nAs mentioned in~\\ref{subsub:write-proc}, Bitcoin blocks are fetched from outside the\nenclave. However, \\mbox{$T^3$}\\xspace verifies the integrity of the Bitcoin block based on the proof of work and the header chain, and\nsince the cost of producing a valid block is expensive, we argue that \\mbox{$T^3$}\\xspace should be able to obtain valid block from the Bitcoin network.\nAlso, \n\\mbox{$T^3$}\\xspace maintains a Merkle Hash Tree (MHT) of the ORAM trees and therefore prevents malicious tampering by verifying all encrypted data fetched from the untrusted memory using the MHT.\nAll encrypted data fetched from the untrusted memory is verified using\nthe MHT.\n\n\\paragraph{Claim 5. The multiple threads involved do not create synchronization issues.}\nHere, it is worth-noting that multiple threads are only involved while\naccessing the Read Tree of \\mbox{$T^3$}\\xspace.\nThanks to the optimized read operation, \\mbox{$T^3$}\\xspace does not run into synchronization\nbugs since there is no memory region that could be simultaneously written to\nby more than one thread.\nIn particular, each thread shares the position map but only reads from the\nposition map.\nEach thread contains its own stash memory which is written to separately by\neach thread.\n\n\\paragraph{Claim 6. The memory interactions within the enclave are\nside-channel-resistant.}\nThe design of $T^3$ incorporates defenses against the side-channel\nthreats~\\cite{Xu15ControlledChannel,hid-sgx-sidechannel-usenix17,shadow-branch-lee-usenix17}\nplaguing Intel SGX.\nIn particular, we used ORAM operations to hide all data access patterns on the untrusted memory region, \nand we incorporated similar oblivious operation techniques introduced in~\\cite{racoon,ndss-AhmadKSL18,\nSasyGF18-zero-trace} to prevent operations inside the enclave from leaking sensitive information.\nFinally, the implementation of \\mbox{$T^3$}\\xspace is also secure against branch-prediction attacks since each\nindividual operation (e.g., accessing Read Tree, updating Write Tree etc.) takes the same sequence\nof branches and therefore reveals no information to the attacker, from the accessed branches.\n\n\\subsection{Denial of Service Attacks from Malicious Clients}\n\nWhile the design of \\mbox{$T^3$}\\xspace is pratical, a malicious client can still incur a large processing time on the server by creating lots of addresses and sending large number of requests for those requests. \nOne way to mitigate such attack is to apply fees on users of the service.\nAnother approach to mitigate denial of service attack is to use a cuckoo filter~\\cite{cuckoo-filter-Fan} to load all addresses from the UTXO set. Upon receiving requests from client, the managing enclave can verify if the address matches the filter as well as the proof of ownership of that address before performing ORAM accesses. Moreover, since Cuckoo filter data structure supports deletion operation, the system can add and remove addresses when performs updating. \nIn other word, in order to perform the denial of service attack, clients need both the proof of ownership as well as a certain amount of Bitcoin in each address. \nHence, it will cost more for the client to perform such attack.\n\n\\subsection{Other Goals Achieved by \\mbox{$T^3$}\\xspace}\nIn this subsection, in addition to the \\textbf{Privacy} goal describe in \\cref{sub:privacy}, we explain how \\mbox{$T^3$}\\xspace achieves the other goals mentioned in\n\\autoref{sub:goal}.\n\n\\paragraph{Validity.} Under the assumption that the adversary does not have enough\ncomputational power to form a new Bitcoin block, the system will only obtain valid \ntransaction by verifying the Merkle root and the proof of work of the Bitcoin block.\n\n\\paragraph{Completeness.} By offering different ways of mapping between Bitcoin addresses and ORAM block id, we can offer services to $92-96\\%$ of all clients\nwith some trade-off between storage overhead and performance.\n\n\\paragraph{Efficiency.} Our contribution to efficiency is threefold. First, our \nsystem is able to handle bursty requests from client concurrently. The core\nidea is to separate the effect of a standard ORAM access into different enclaves.\nThus, the multiple reading enclave can concurrently perform read operations at\nthe same time that the writing enclave can perform a non-blocking Evict \nprocedure on the other tree. Second, by having two ORAM trees, we minimize the\ndowntime of the system by having the writing enclave performed updates on one\ntree and reading enclave handled clients' requests on the other tree. \nThe server downtime depends on the number of requests that the system \nreceives when the writing enclave performs ORAM updates on the original ORAM \ntree. Finally, by enforcing clients to provide the proof of ownership of\nthe address and assuming that a honest client is rational, we prevent other clients from querying addresses that do not belong to them.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Design Goals and Solution Overview}\n\\label{sec:Overview}\nIn this section, we define the system components, outline our security goals, and give an overview of how our system works.\n\\subsection{System Components}\n\\label{sub:system_components}\nThere are three key components of this system: the Bitcoin network, a client, and a untrusted server.\n\tThe \\textbf{Bitcoin network} is a set of nodes that maintains the Bitcoin blockchain, and the network validates and relays the new Bitcoin block produced by miners. \n\n\tA \\textbf{client} is a Bitcoin simplified payment verification node that remotely connects to the secure TEE on the untrusted server to perform oblivious searches on the unspent transaction output (UTXO) set. The client is also able to connect to the Bitcoin Network to obtain other network metadata such as the latest Bitcoin block header. \n\n\tA \\textbf{server} is the untrusted entity made up of two components: an untrusted server and several trusted TEEs (i.e., the \\textit{managing}, \\textit{reading}, and \\textit{writing} TEEs). Moreover, the untrusted server stores three encrypted databases which are the \\textit{read-once} ORAM tree\\xspace, the \\textit{original} ORAM tree\\xspace, and the Bitcoin header chain. The untrusted server hosts a potentially malicious bitcoin client (e.g., $\\mathsf{bitcoind}$) that handles the communication with the Bitcoin Network. \n\n\n\\subsection{Design Goals}\n\\label{sub:goal}\nThe goal of our system is to leverage the trusted execution capabilities of Trusted Execution Environment (TEE) with attestation to design a public Bitcoin full node that supports oblivious search and update on the current Bitcoin unspent transaction output database. \nOur system aims to provide data confidentiality and privacy to Bitcoin SPV clients in a large scale by using standard encryption and Oblivious RAM techniques on the current set of unspent transaction outputs. \nThe main goals that \\mbox{$T^3$}\\xspace tries to achieve are:\n\\begin{compactenum}\n\t\\item \\textbf{Privacy.} \\mbox{$T^3$}\\xspace aims to provide privacy and confidentiality to SPV clients' requests. \n\t In particular, \n\t \n\t the system allows SPV clients to obliviously search its relevant transactions without revealing their addresses to potentially malicious providers by using TEE to encrypt the data and using ORAM schemes to eliminate known side channel leakages~\\cite{racoon,ndss-AhmadKSL18,thang-hoang:posup-popets,SasyGF18-zero-trace}. \n\t\\item \\textbf{Validity.} In our design, the SPV client should be able to obtain valid information based on the provided addresses, \n\t and a malicious adversary should not able to tamper the Blockchain data with invalid transaction outputs. \n\t\\item \\textbf{Completeness.} The system should provide clients with access to most of its relevant transactions in order to determine balance or to obtain essential information to form new transactions.\n\t\\item \\textbf{Efficiency.} The system should be practical to deploy. \n\t\t\t\t\t\tMore specifically, the system should be efficient enough to handle different concurrent SPV clients' requests without compromising the privacy of the clients. \n\n\\end{compactenum}\n\n\n\n\n\n\\subsection{Solution Overview}\n\tThe idea of using ORAM schemes and trusted execution environment to construct database systems that support oblivious accesses has been investigated by the research community~\\cite{thang-hoang:posup-popets,oblidb,SasyGF18-zero-trace}.\n\tHowever, the efficiency and scalability of those systems are hampered by the lack of concurrency of traditional ORAM schemes~\\cite{Stefanov:2013,wang-circuit-oram-2015}.\n\n\tIn this work, we design \\mbox{$T^3$}\\xspace to overcome the limitations of efficiency and\n\tconcurrency plaguing existing systems. Our design is motivated by\n\tthe following observations. \n\tThe first observation is that each ORAM access in a standard tree-based ORAM settings is a combination of two operations: a \\textit{read-path} operation and an \\textit{eviction} operation.\n\tBy separating the effects two operations into two different trees: a \\textit{read-once} ORAM tree\\xspace and an \\textit{original} ORAM tree\\xspace, one can use read-path operation on the \\textit{read-once} ORAM tree\\xspace to handle clients' requests simultaneously while performing a non-blocking eviction operation on the \\textit{original} ORAM tree\\xspace sequentially.\n\tThis design is also independently investigated by ConcurORAM~\\cite{concurORAM-chakraborti}; however, their design is not suited\n\tfor TEEs with limited trusted memory (such as Intel SGX). We elaborate on this\n\tin the coming sections.\n\t\n\tThe second observation is that the access privacy guarantee of this approach relies the characteristic of the Bitcoin blockchain. \n\tIn particular, the Bitcoin network generates new Bitcoin block on average of 10 minutes, and if we require \\mbox{$T^3$}\\xspace to periodically synchronize these the two trees, then the privacy of clients' queries are preserved. \n\n\tMoreover, if we assume that upon receiving transactions belonged to its addresses, the rational client should not query same transactions {\\em again} until the next block arrives, the proposed approach on the separation of \\textit{read-path} and \\textit{eviction} procedure not only does not affect the privacy guarantees of ORAM access but also allows \\mbox{$T^3$}\\xspace to handle much more clients' requests. \n\tMore importantly, we also argue that even when the SPV clients are irrational by submitting requests for the same transaction more than one, \n\tthe privacy of those clients is only compromised for a short period of time (i.e., 10 minutes for the Bitcoin network) because \\mbox{$T^3$}\\xspace will always synchronize the old instance of the \\textit{read-once} ORAM tree\\xspace with the more updated instance of the \\textit{original} ORAM tree\\xspace.\n\tWith the intuition of \\mbox{$T^3$}\\xspace described above, we outline the workflow\n\tof our design:\n\n\n\\noindent\t\\textbf{Server Initialization}~\\rcircled{1}-\\rcircled{8}: \n\n\n\n\n\tInitially, the managing TEE will initialize a \\textit{writing} TEE\\xspace that creates an empty ORAM tree. \n\tFor each of Bitcoin block obtained from the network, the managing TEE verifies the proof of work of the block before passing relevant update data to the \\textit{writing} TEE\\xspace in order to populate the ORAM tree.\n\n\tWith the current size of the Bitcoin blockchain, this operation might take several hours. \n\tHowever, once the TEEs catch up with the current state of the Bitcoin blockchain, we expect that the TEE only has to perform a batch of update accesses on the ORAM tree every 10 minute.\n\tWhen the initialization is completed, the \\textit{managing} TEE\\xspace creates two copies of the ORAM tree which are the \\textit{read-once} ORAM tree\\xspace and the \\textit{original} ORAM tree\\xspace. \n\n\n\n\\noindent\\textbf{Oblivious \\textit{\\textit{read-once}}\\xspace Protocol}~\\bcircled{1}-\\bcircled{6}: In order to obtain its unspent outputs, the client first performs the remote attestation to the \\textit{managing} TEE\\xspace. \n\tThe remote attestation mechanism allows the client to verify the correctness of program execution inside the TEE. \n\tMore importantly, after a successful attestation, the client can use standard key exchange mechanism (i.e. Diffie-Hellman's key exchange) to share a secret session key with the TEE in order to establish a secure connection with the \\textit{managing} TEE\\xspace. Upon receiving client's connection requests, the \\textit{managing} TEE\\xspace creates a \\textit{reading} TEE\\xspace with its own copies of the ORAM position map and the ORAM stash to handle client subsequent requests. \n\n\tNext, after having a secure channel, the client will send his Bitcoin addresses along with the proof of ownership of those addresses to the TEE (e.g., the knowledge of the public key along with a signature to a random nonce or the preimage of the public key hash). \n\tThe \\textit{reading} TEE\\xspace will use a mapping function to map Bitcoin addresses into the ORAM block identification number and performs \\textit{read-once} ORAM access\\xspace on the ORAM tree. \n\tIn particular, those \\textit{read-once} ORAM access\\xspace do {\\em not} involve the eviction procedure which requires re-encrypting and remapping the ORAM block. \n\tThe eviction procedure will be performed on the \\textit{original} ORAM tree\\xspace~by the \\textit{writing} TEE\\xspace.\n\n\n\n\n\n\n\t\n \n \n\\noindent \\textbf{Oblivious Write Protocol}~\\rcircled{1}-\\rcircled{8}: \n\tThe \\mbox{$T^3$}\\xspace requires to update the ORAM tree via batch of write accesses every 10 minutes on average. \n\tIn particular, \n\n\t\\mbox{$T^3$}\\xspace will rely on a standard Bitcoin client \n\tto handle the communication with the Bitcoin network to obtain blockchain data\\footnote{This ability can be easily included in the future implentation of \\mbox{$T^3$}\\xspace.}.\n\n\n\n\n\tThus, \\mbox{$T^3$}\\xspace needs to verify the block relayed by a potentially malicious Bitcoin client before updating the ORAM tree. \n\tMore specifically, in the design, \\mbox{$T^3$}\\xspace stores a separate Bitcoin header chain to verify the proof of work and the validity of all transactions inside a Bitcoin block.\n \tAfter the verification, the \\textit{managing} TEE\\xspace forms a batch of ORAM updates and delegates those updates to the \\textit{writing} TEE\\xspace. \n\tOnce those updates are finished, the \\textit{managing} TEE\\xspace will queue up read requests from SPV clients in order to allow the \\textit{writing} TEE\\xspace to finish the eviction requests from the $\\textit{read}$ TEEs during the updating interval. \n\tAs soon as the \\textit{writing} TEE\\xspace finishes performing those eviction requests, \n\tthe \\textit{managing} TEE\\xspace updates the position map and \\textit{stash}, and makes the ORAM tree used by the \\textit{writing} TEE\\xspace become the new ORAM tree used by \\textit{reading} TEE\\xspace.\n\tAt this point, the \\textit{reading} TEE\\xspace can use the new tree instance to respond to clients' requests while the \\textit{writing} TEE\\xspace performs the eviction procedure on another copy of the same ORAM tree.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION}\n\nNeutron stars in low-mass X-ray binaries (LMXBs) accrete matter\nfrom a companion with a mass of less than $1\\,\\rm{M_{\\sun}}$ via\nan accretion disk. In many models, the Keplerian accretion disk is\nsupposed to be terminated at an inner radius $r_{\\rm in}$ of a few\nSchwarzschild radii by e.g. relativistic effects, radiation drag,\nor neutron-star magnetosphere-disk interactions (see van der Klis\n2004 for a review). The motion of the inner disk flow and the\ngeometry of the innermost disk are still uncertain. The\nobservations of the correlated behaviors in X-ray spectral and\ntiming variabilities in LMXBs have provided the important probes\nof the accretion-flow dynamics and the stellar-disk interactions.\n\nKilohertz quasi-periodic oscillations (kHz QPOs, in the frequency\nrange of 200 Hz--1300 Hz) have been observed in more than 20\naccreting neutron star LMXBs (see van der Klis 2000, 2004 for the\nrecent reviews). Their frequencies are usually regarded to be\nassociated with the Keplerian orbital frequency at some preferred\nradius related to $r_{\\rm in}$. An important evidence for the\nmovement of the inner edge of the disk under different mass\naccretion rate comes from the observation of a positive\ncorrelation of frequency vs. count rate on timescales of hours to\ndays in some low-luminosity LMXBs, i.e. ``atoll\" sources\n\\citep{mend99}. A similar correlation (or in some cases an\nanti-correlation) has been also found in some high-luminosity\nLMXBs, i.e. ``Z\" sources, in the form of the correlations with\ncurve length $S_{\\rm z}$ along the track traced in an X-ray\ncolor-color diagram \\citep{wijn97,yu01,homan02}. The effect of\nstellar radiations on the disk movement has also been observed,\ne.g. in 4U 1608--52, where \\citet{yu02} found the\nanti-correlations between the kHz QPO frequency and the X-ray\ncount rate associated with mHz QPO due to nuclear burning on the\nneutron star surface.\n\nNearly coherent brightness oscillations have been discovered\nduring thermonuclear X-ray bursts in some LMXBs (see Strohmayer \\&\nBildsten 2003 for a review). Recent observations of the accreting\nmillisecond pulsars, SAX J1808.4--3658 \\citep{chak03} and XTE\nJ1814--338 \\citep{mss03}, confirm that the burst oscillation\nfrequency is extremely close to the spin frequency. It supports\nthe interpretation of the burst oscillation in terms of a ``hot\nspot\" on the stellar surface \\citep{chak03,SB03}, which\n triggers the efforts to investigate X-ray emission properties\non the neutron star surface by folded pulse profiles\n\\citep{stro03,bhat05,watt05}. The best-fitting parameters of the\npulsed fraction and the first harmonic can not only determine the\ncompactness of neutron star, but also help to figure out how the\nnuclear burning front (the hot spot) propagates on the stellar\nsurface \\citep{bhat05,watt05}. However, in the burst-oscillation\nstudies, because of the rather limited photon statistics on the\nshort burst timescale, many similar bursts have to be added to\nimprove the statistics of the folded pulse profiles.\n The underlying problem is that these bursts\nmay occur in different mass accretion rates. Till now, we know\nlittle about what effects of the accretion rate fluctuations can\nmake on the neutron-star surface emissions or material\ndepositions.\n\nThis situation can be changed with the discovery of the accreting\nmillisecond X-ray pulsars (MXPs, see Wijnands 2005 for a latest\nreview). Their detectable coherent pulsations allow us to get the\npulse profile with good statistics over a long observational\nperiod when the energy spectrum (or hardness ratios) varies only\nslightly. According to the detailed spectroscopic and\npulse-profile analysis on SAX J1808.4--3658, \\citet{pout03}\nconstructed a model for X-ray emissions from a hot spot on the\nsurface\/boundary layer of a rapidly rotating neutron star in the\nhard island state. They used it to constrain the compactness of\nthe central star, the position of the emitting region and the size\nof the hot spot. Their investigations shed a light on further\nstudies of the variations of the surface X-ray emission regions\nduring the evolution of the accretion disk.\n\nAmong the discovered 7 millisecond X-ray pulsars, XTE J1807--294\nis the best candidate for us to investigate the impact of disk\nflow activities on the neutron-star surface emission for four\nreasons: (1) Besides SAX J1808.4--3658, XTE J1807--294 is also a\nsource which has been reported to have twin kHz QPOs \\footnote{In\ncompletion of our paper, we noted that Linares et al. (2005) used\nthe same data of\n XTE J1807--294 as we analyzed. Our results are similar\n in kHz QPOs and different in the low frequency ranges, which might be due to\n the different choices of the data segments and different data grouping.}\n\\citep{wijn03,wijn05,lina05}.\n (2) The binary parameters of it have\nbeen calculated by \\citet{camp03} and \\citet{kirs04} based on the\n{\\it XMM-Newton} observation, so we are able to correct the photon\narrival times for the orbital motions in producing pulse profiles.\n(3) The time-averaged energy spectrum of the source is found to be\ndominated by an optically-thin Comptonized component\n\\citep{fala05}, similar to that of SAX J1808.4--3658\n\\citep{Gier02}, so that we can take the X-ray emission model of\nSAX J1808.4--3658 as a direct reference. (4) There are no\nthermonuclear bursts reported in this source; thus we can focus on\nstudying the effect of disk evolutions. In the left 6 MXPs, i.e.,\nSAX J1808.4--3658, XTE J1751--305, XTE J0929--314, XTE J814--338,\nIGR J00291$+$5934 and HETE J1900.1--2455 (see Wijnands 2005 for a\nreview), the above four conditions cannot be satisfied\nsimultaneously. With the knowledge mentioned above, many basic\nfeatures of XTE J1807--294 have been available as well, such as\nthe shortest orbital period of $\\sim 40$ minutes and a relatively\nslow spin frequency of $\\sim$ 191 Hz \\citep{mark03c}. In addition,\nthis source locates at $5^{\\circ}.7$ away from the Galactic\ncenter, with the best known position reported by \\citet{mark03c}\nbased on a Chandra observation. Assumed a distance of 8 kpc, the\nsource luminosity dropped from $1.3\\times\n10^{37}\\,\\rm{erg\\,s^{-1}}$ on February 28, 2003, to $3.6\\times\n10^{36}\\,\\rm{erg\\, s^{-1}}$ on March 22, 2003 \\citep{fala05}.\n\nWe analyze the evolution of light curves, hardness ratios, pulse\nprofiles and power density spectra of XTE J1807--294, using {\\it\nRossi X-ray Timing Explorer} ({\\it RXTE}) observations from\nFebruary 27 to March 31, 2003. Our results firstly show that the\npositive kHz QPO frequency-count rate correlation on timescales of\nhours to days is related to some ``puny\" flares, originated from\ndisk flow inhomogeneities; the behaviors of the pulsed fraction\nand the first harmonic in the flares are different from those in\nthe non-flares. However, in the whole investigated episode, there\nis a positive correlation between the pulsed fraction and the kHz\nQPO frequency, which could be the first evidence that neutron-star\nsurface emissions are very sensitive to the disk flow\ninhomogeneities. Furthermore, we describe the {\\it RXTE}\nobservations and data analysis method in Section 2; the results\nof timing analysis are presented in Section 3 and discussions of\nthe results are shown in Section 4. Finally, we make a brief\nconclusion of our studies in Section 5.\n\n\n\\section{{\\it RXTE} OBSERVATIONS AND DATA ANALYSIS}\n\nWe study the data obtained by the Proportional Count Array (PCA)\non board {\\it RXTE} during the March 2003 outburst decay of XTE\nJ1807--294, using FTOOLS 5.3.\n\nTo have an overview of the source evolution, we extract light\ncurves from the PCA standard 2 mode data (detector 2 only, all\nlayers), and subtract the background produced by `pcabackest'\nversion 3.0 with the ``CM\" faint-source model. We define soft\ncolor as the ratio of count rates in the $4.1-7.0\\,\\rm{keV}$ and\n$2.0-4.1\\,\\rm{keV}$ bands and hard color as the ratio of count\nrates in the $13.3-30.2\\,\\rm{keV}$ and $7.0-13.3\\,\\rm{keV}$ bands.\nIn selecting good time intervals for data analysis, we give a\ncondition of PCU2$_{-}$on.eq.1 in addition to the standard\nconditions for faint sources. Then we extract the light curves of\nPCU 0, 1, 3, 4 separately and compare them with the light curve of\nPCU2 respectively. Finally, we find the time intervals when PCU 4\nmeet instrumental flares and delete such bad time intervals,\nmaking sure the following analysis are not influenced by\ninstrumental problems.\n\nFor Fourier analysis, we construct pulse profiles and power\ndensity spectra from the $\\sim 0.95\\,\\mu$s time-resolution PCA\nGoodXenon mode data (all available detectors, all layers),\ncorrecting the photon arrival times for orbital motions of the\npulsar and the spacecraft. When dealing with the GoodXenon mode\ndata, we do not perform background subtraction or deadtime\ncorrection.\n\nFor pulse-profile fitting we create folded light curves in 32\nphase bins for every continuous event file. As a fit function of\nthe pulse profiles we adopt a two-component Fourier function of\nthe form\n\\begin{equation}\n I(\\phi)=c_0(1+a_0\\sin\n[2\\pi(\\phi-\\phi_0)]+a_1\\sin [2\\pi(\\phi-\\phi_1)\/0.5])\n\\end{equation}\nwhere $c_0$ is the normalized count rate of the pulse profile;\n$a_0$ and $a_1$ are the fractional amplitude of the fundamental\nand the first harmonic component, respectively; $\\phi_0$ and\n$\\phi_1$ are the corresponding zero phase angles. We fit the pulse\nprofiles and determine the errors of the fit parameters by using\n$\\Delta \\chi^2=2.7$. We estimate the real averaged count rate from\nthe $2-60\\,\\rm{keV}$ net light curve, produced by processing\nStandard 2 mode data (detector 2, all layers) within the\ncorresponding intervals of every folded pulse profile.\n\nWe construct power density spectrum (PDS) per observation using\nthe data segments of 256 s and 1\/4096 s time bins, so that the\nlowest available frequency is 1\/256 Hz and the Nyquist frequency\nis 2048 Hz. We normalize the power spectra by the method of\n\\citet{miya91}, and subtract the Possion white noise. In order to\nimprove the statistics, we add the observations into groups when\nthese observations are adjacent in time and the power spectra\nremain the same. The division of the groups can be found in Table\n\\ref{tab_observation} and are depicted in the top panel of Fig.\n\\ref{fig_outburst}. Most of our power spectra show a\npower-law-like component in mHz--Hz frequency range, which could\nbe the very low frequency noise (VLFN). Furthermore, in our PDS,\nsome additional weak noises can be measured with good statistics\nas shown in Section 3.3.\n\nWe use a power law plus multi--Lorentzian function\n\\citep{bpk02,vstr02} to fit the spectra. Before fitting the power\nspectra, we remove frequency bins containing the pulse spike. To\nvisualize the characteristic frequencies ($\\nu_{\\rm max}$) of\nnoises or QPOs, we plot the power spectra in the power times\nfrequency representation ($\\nu{\\rm P}_{\\nu}$), where ${\\rm\nP}_{\\nu}$ is the power spectral density and $\\nu$ is the Fourier\nfrequency. In this representation, the Lorentzian's maximum occurs\nat $\\nu_{\\rm max}$ ($\\nu_{\\rm max} = \\sqrt{\\nu_0^2 + \\Delta^2}$,\nwhere $\\nu_{\\rm max}$ is the characteristic frequency, $\\nu_0$ is\nthe centroid frequency and $\\Delta$ is the half width and half\nmagnitude (HWHM) of the Lorentzian). We represent the Lorentzian\nrelative width by $Q$ defined as $\\nu_0\/2\\Delta$, and the\nroot-mean-square fractional amplitude by $rms$.\n\n\n\\section{RESULTS}\n\n\\subsection{Broad ``Puny\" Flares in 2003 March }\n\nIt has been found that the light curve of XTE J1807--294 drops\nexponentially from February to the end of March of 2003\n\\citep{fala05} and then becomes flattened instead of being much\nsteepened as in SAX J1808.4-3658 \\citep{Gilf98,Gier02} and XTE\nJ1751--305 \\citep{GP05}. Figure \\ref{fig_outburst} shows that in\nthe former exponentially decaying episode, at least four intensity\nfluctuations occur, each lasting beyond several thousands of\nseconds, with small X-ray count rate variations. To make\nquantitative analysis, we fit the light curve by an exponential\nplus multi-Gaussian model. The full width and half magnitude\n(FWHM) of each Gaussian and the ratio of the magnitude of the\nGaussian to the exponential value at the same time are\n($0.53\\,\\rm{day}$, $7.18\/45.42$), ($0.95\\,\\rm{day}$,\n$16.8\/35.22$), ($1.67\\,\\rm{day}$, $7.53\/20.75$),\n($0.71\\,\\rm{day}$, $12.33\/17.16$), respectively. The features of\nlong duration and low amplitude make these fluctuations different\nfrom type I thermonuclear bursts and super nuclear outburst\n\\citep{SB02, SM02} on the neutron star surface. We thus call them\n``puny\" flares for convenience.\n In the following, it can be\nnoticed that these ``puny\" flares are also featured by the\n enhanced soft X-ray emissions and increased pulsed fractions.\n\nThe second panel of Fig. \\ref{fig_outburst} shows that the soft\ncolor drops in the flares, especially at the top of the largest\nflare of Mach 14--15. By comparing the light curves in the four\nenergy bands ($2.0-4.1\\,\\rm{keV}$, $4.1-7.0\\,\\rm{keV}$,\n$7.0-13.3\\,\\rm{keV}$ and $13.3-30.2\\,\\rm{keV}$), we find that this\nis due to the stronger soft X-ray emissions in the band of\n$2.0-4.1\\,\\rm{keV}$. The color-intensity diagram of Fig.\n\\ref{fig_J1807_hid} indicates that, except of the ``puny\" flares,\nthe soft color decreases with the decay of the outburst, whereas\nthe hard color changes a little in the whole investigated episode.\n\nWe can deduce from the almost invariant hard color that the hard\nspectral component changes only slightly in the whole episode. The\nfitting results of the combined spectra of {\\it XMM-Newton}, {\\it\nRXTE} and {\\it INTEGRAL} observations \\citep{fala05} present that\nhard emissions from the hot and optically thin Comptonized plasma\ndominate the energy spectra; the contribution of thermal emissions\nfrom the accretion disk at February 28 and March 22 are so small\nthat they are almost negligible in PCA analysis (above 2.5 keV).\nTherefore, the PCA X-ray emissions can be considered to be mainly\ngenerated from the neutron star surface\/boundary layer.\n\n\\subsection{Variations of Pulse Profiles}\n\nWe fit 86 pulse profiles with a two-component sine function\ndescribed in Section 2. The fitting is good in most cases, with\n$\\chi^2\/\\rm{dof}$ ranged in $32\/59-80\/59$. The situations of\n$\\chi^2\/\\rm{dof}>1.3$ only occur in three pulse profiles. We plot\nthe evolutions of the best-fitting parameters of $a_0$ and $a_1$\nin the fourth and the fifth panel of Fig. \\ref{fig_outburst},\nrespectively. It can be seen that $a_0$ is a rather unambiguous\nindicator of the ``puny\" flares than the X-ray intensities and\nsoft colors, not only because of its small errors, but also\nbecause $a_0$ increases significantly at the top of the flares\ndespite how weak the flare is.\n\nFigure \\ref{fig_J1807_psfile} shows the different behaviors of\n$a_0$ and $a_1$ with respect to the averaged count rate of the\npulse profile. Combining the fourth panel of Fig.\n\\ref{fig_outburst}, we find that in the normal state, the first\nharmonic is relatively strong, but it becomes weaker (with $a_1$\ndecreasing from $1.4\\%$ to about $0.9\\%$) in the decay of the\noutburst, whereas $a_0$ keeps at $\\sim 6.5\\%$; in the flares,\n$a_0$ is positively correlated with the averaged count rate,\nwhereas $a_1$ is generally in a lower level around $0.7\\%$ over\nthe flares. The variational range of $a_0$ is $\\sim 2\\% - 14\\%$.\n\n\n\\subsection{Aperiodic Timing Behavior}\n\nWe fit the power density spectra by one power-law component plus\nthree to seven Lorentzian components. Errors on the fit parameters\nare determined using $\\Delta \\chi^2=1$. We list the characteristic\nparameters of the Lorentzians, e.g. $\\nu_{max}$, $Q$ and $rms$ in\nTable 2, the power-law index $\\alpha$ of the VLFN and the\n$\\chi^2\/{\\rm dof}$ of the fitting in Table 3. In our investigated\nepisode, kHz QPOs have been detected in groups 1-7 and 9-10, and\nundetectable in the other groups on account of the poor statistics\nof the power spectra above $\\sim$10 Hz. The typical power spectra\nof XTE J1807--294 are shown in Fig. \\ref{fig_pdsfit}, where group\n1 is chosen to stand for the non-flare state and group 5 is taken\nas a representation of the flare state.\n\nWe plot the characteristic frequencies of the Lorentzians versus\nthe upper kHz QPO frequency in Fig. \\ref{fig_relation}. It shows\nthat XTE J1807--294 has a frequency-frequency correlation similar\nto that of SAX J1808.4--3658 \\citep{vstr05}. Most of the\nLorentzian components can then be identified in the scheme of\n\\citet{bpk02} and \\citet{vstr03}. In Fig. \\ref{fig_relation},\nL$_u$ is the upper kHz QPO; L$_\\ell$ is the lower kHz QPO;\nL$_{hHz}$ is the hectohertz Lorentzian, a broad Lorentzian with a\nfrequency around 150 Hz; L$_{b}$ is the break Lorentzian, a\nband--limited noise component; L$_h$ is the low frequency\nLorentzian just above L$_{b}$. It should be noted that when the\nfrequency of L$_\\ell$ approaches the range of $100\\sim 150$ Hz, it\nbecomes difficult to be distinguished from L$_{hHz}$, especially\nwhen one of them is not detected in the spectra, e.g. groups 3 and\n4 (see Fig. \\ref{fig_relation}). In Table 2 we put `?' on their\nidentifications.\n\nIn Fig. \\ref{fig_relation} we multiply the frequencies of twin kHz\nQPOs of Z sources GX 5$-$1 and GX 17$+2$ by a factor of 1\/1.5 for\ncomparison. Their shifted frequencies are almost consistent with\nthe frequency--frequency distribution of the twin kHz QPOs of XTE\nJ1807--294. We draw $Q$ versus $\\nu_{\\rm max}$ for both L$_u$ and\nL$_h$ in Fig. \\ref{fig_qvsnu}. It proves that the frequency shift\nexists in the upper kHz QPO frequencies in XTE J1807--294 as in\nthe case of SAX J1808.4--3658 \\citep{vstr05}.\n\nExcept of the five major Lorentizans of L$_u$, L$_\\ell$,\nL$_{hHz}$, L$_h$ and L$_{b}$, the power spectra of XTE J1807--294\npresent some additional components (Fig. \\ref{fig_relation}).\nOne of the components is the Lorentizan of L$_{h2}$, with\nfrequency nearly double of L$_{h}$. If L$_{h}$ corresponds to the\nHBO component of Z sources (see van der Klis 2004 for a review),\nL$_{h2}$ would correspond to the HBO harmonic. Such harmonic has\nnever been reported in atoll sources or other accreting\nmillisecond pulsars. Another additional component is the\nLorentizan of L$_{LFN}$, which takes the position of the shifted\nlow-frequency noise (LFN) component of GX$17+2$ in the diagram\n(Fig. \\ref{fig_relation}). The fitted power law index of\n$\\nu_{LFN}$ vs. $\\nu_{u}$ is $3.42\\pm 0.29$, larger than the index\nof $\\nu_{b}$ vs. $\\nu_{u}$ of $\\sim$2.7. We use 'Ftest' to check\nthe probability of the components whose errors are less than\n$2\\sigma$. All of the weak noise components listed in Table. 2\nhave Ftest probabilities less than $0.03$.\n\nThe simultaneous detection of L$_{b}$ and L$_{LFN}$ in XTE\nJ1807--294 makes it evident that L$_{LFN}$ is a different\ncomponent from L$_{b}$. We infer from this finding that the\ndetected LFN in GX $17+2$ \\citep{homan02} should be a different\ncomponent from those observed in other Z sources. The shift factor\nof about 1.5 in L$_{LFN}$ as well as in L$_u$ and L$_{\\ell}$ imply\nthat these three components could originate from the same inner\ndisk region.\n\n\\subsection{Correlations Between Count Rate, Pulse Profile and kHz QPO }\n\nWe draw in Fig. \\ref{fig_a0QPO} the upper kHz QPO frequency\n($\\nu_{\\rm u}$) versus the averaged count rate and the fractional\npulse amplitude ($a_0$) of the group, respectively. The averaged\ncount rate and $a_0$ of every group were calculated using the same\nmethods described in section 3.2.\n\nWe can see from Fig. \\ref{fig_a0QPO} and Fig. \\ref{fig_relation}\nthat, while the frequencies of the Lorentizans decrease with the\ncount rate in the non-flare state of groups 1--3 and increase with\nflare intensities in groups 4--6 and 7--10, they do not vary\nmonotonously with the X-ray count rate in the whole episode. The\npositive frequency--count rate correlations in the rise of the\nflares seem to be steeper than that measured in the normal\noutburst decaying state. However, in the whole investigated\nepisode there exists a noticeable positive $\\nu_{\\rm u}$ vs. $a_0$\nrelation with a linear slop coefficient of $\\sim 35$ (Fig.\n\\ref{fig_a0QPO}).\n\n\\section{DISCUSSION}\n\nOur motivation to do timing analysis on XTE J1807--294 is to probe\nthe effects on the neutron-star surface X-ray emissions made by\nthe disk flow activities, so we focus on analyzing the variations\nof pulse profiles and power density spectra during the evolution.\n\nWe find several pairs of twin kHz QPOs in the power density\nspectra of XTE J1807--294. The frequency separation $\\Delta \\nu$\nranges from $\\sim 179\\,\\rm{Hz}$ to $\\sim 247\\,\\rm{Hz}$, close to\nthe spin frequency of $\\nu_{\\rm s}\\sim 191\\,\\rm{Hz}$. Several kHz\nQPO models, considering the coupling of the neutron-star spin to\nthe Keplerian orbit motion of material in the accretion disk, have\nbeen put forward to explain this specific feature. We notice that\nmost of these models concern inhomogeneous accretion flow by for\nexample accreting blobs, density fluctuations in the innermost\npart of the accretion disk, or high-density loops confined along a\nmagnetic field line. In the modified sonic-point beat-frequency\nmodel \\citep{lamb01,lamb03}, the accreting blobs are introduced to\nexplain the deviation of $\\Delta \\nu$ from the constant value of\nthe spin frequency by considering their relative motions to the\nneutron star. In the resonant disk oscillation models\n\\citep{kluz04,lee04}, the twin kHz QPOs are explained by a\nnonlinear resonance excited by coupling disk oscillation modes to\nthe neutron star spin. \\citet{petr05} proved further that a\nrotating misaligned magnetic field of a neutron star or a rotating\nnon-axially symmetric gravitational potential can provide the\ncoupling; their perturbations on the accretion disk can give rise\nto very pronounced density fluctuations at the inner edge of the\ndisk. Most recently, \\citet{li05} pointed out an alternative\ninterpretation for the lower kHz QPO in terms of the standing kink\nmodes of magnetic loop oscillations at the inner edge of the disk,\nwhere the loops with high-density plasma and small cross section\ncan be produced from the reconnection of the azimuthal magnetic\nfield lines.\n\nThe variation of kHz QPO frequencies can be explained by the\nmovement of the inner edge of the accretion disk in the radiative\ndisk truncation model \\citep{mill98,klis01}. For example, the\nobserved positive frequency-count rate correlations on timescales\nof hours to days in some atoll sources have been interpreted as\nthe changes in the inner disk radius due to disk accretion rate\nfluctuations. We find a similar frequency-count rate correlation\nin XTE J1807--294, but we notice that the shift of the QPO\nfrequency is related to some broad ``puny\" flares. The observed\nfrequency-count rate correlation in the rise of the flares seems\nto be steeper than in the normal state. This result means that (1)\nin the case of the same count rate, the kHz QPO frequency measured\nin the rise of the flare is larger than that in the normal state;\n(2) for a certain kHz QPO frequency range, the variation of the\ncount rate is smaller in the flare than in the normal state. A\npossible explanation is that there are blobs of high mass\ndensities at the inner edge of the accretion disk and the they can\npersist in time scales of hours to days. The formation and the\ndetailed structure of such a kind of blob are still uncertain. We\nrefer to the hydrodynamic simulation result by \\citet{petr05}\nmentioned in the above paragraph for instance. It might be\nexpected that the relative intensity of the flares to the nearby\nnormal state could be a measure of the mass density of the blob;\nwhile the limited time intervals of the flares could be related to\nthe size of the blobs. However, besides the variation of mass\naccretion rate due to accreting the blobs, there are other\nmechanisms which can influence the X-ray intensity and kHz QPO\nfrequency, such as nuclear burning on the neutron star surface\n\\citep{bild93,yu02}. To make a further quantitative investigation\nof the blobs from the observation, we have to distinguish their\ndifferent effects firstly.\n\nAccreting inhomogeneous flow as one of the possible origins of\nflares has been suggested by \\citet{moon03} for LMC X--4. In the\n``broad flare\" of LMC X--4, the spectrum is softened and the pulse\nprofile becomes simple sinusoids. These features are also observed\nin the broad ``puny\" flare of XTE J1807--294. In the massive X-ray\nbinary LMC X--4, the X-ray emission mechanisms of the magnetic\nneutron star (with surface magnetic field strength $B\\sim\n10^{12}\\,\\rm{G}$) are too complex to be modelled. However, the\nspectrum of XTE J1807--294 can be well fitted by an absorbed\nblack-body plus Comptonization model; thus we can hope to learn\nmore about the X-ray emission properties by fitting the pulse\nprofiles of XTE J1807--294 as done by \\citet{Gier02,GP05} for SAX\nJ1808.4--3658 and XTE J1751--305.\n\nOur preliminary results of the pulse-profile analysis present that\nboth $a_0$ and $a_1$ behave differently in the flares and in the\nnon-flares. As for the latter case, while the X-ray count rate\ndecreases, $a_0$ keeps at about $6.5\\%$, whereas $a_1$ decreases\nfrom $\\sim 1.4\\%$ to $\\sim 0.9\\%$. However, in the former case,\n$a_1$ is generally in a low level of $0.7\\%$, whereas $a_0$\nincreases with the flare intensities. The value of $a_0$ spans in\nthe range of $\\sim 2\\% - 14\\%$, comparable to those observed in\nsome thermonuclear bursts of XTE J1814--338 \\citep{stro03,watt05}\nand some non-pulsing LMXBs \\citep{muno02}. According to the X-ray\nemission model of \\citet{pout03}, the variation of the pulse\nprofile can be determined by the changes of the surface-emission\nparameters, such as the hot spot size, position, and even emission\npatterns, e.g. the optical depths of the black-body emission and\nthe Comptonization, respectively. We will put the detailed\nspectroscopic and pulse-profile modelling analysis in a future\nwork. According to this model, the above results have already\nshown that the accretion emissions at the neutron star\nsurface\/boundary-layer are very sensitive to the accretion flow\ninhomogeneities. The most direct evidence is the observed positive\ncorrelation between kHz QPO frequency and $a_0$ in the\ninvestigated episode.\n\nAlthough we infer from the above results that the broad ``puny\"\nflares could be mainly due to accreting blobs, we cannot exclude\nthe possibility of the existence of some special mode of nuclear\nburning on the neutron star surface. \\citet{bild93} firstly\nproposed that a time-dependent nuclear burning (slow fires) on\npatches of the neutron star surface can occur in an intermediate\naccretion regime, where the accretion rate is sub-Eddington but is\nstill too high to allow type I thermonuclear burst, e.g. $5\\times\n10^{-10}\\,\\rm{M_{\\sun}yr^{-1}}<\\dot{M}<10^{-8}\\,\\rm{M_{\\sun}yr^{-1}}$\nfor a pure helium burning case \\citep{bild95}. For XTE J1807--294,\nthe conditions for slow fires seem to be satisfied qualitatively:\nno type I thermonuclear burst has been observed and the source\nluminosity drops from $1.3\\times 10^{37}\\,\\rm{erg\\,s}^{-1}$ to\nabout $3.6\\times 10^{36}\\,\\rm{erg\\,s}^{-1}$ (assuming the distance\nof 8 kpc) in the investigated episode \\citep{fala05}. However, the\nproperty of the companion star is still uncertain. It could be a\nHe dwarf with a low inclination angle ($i<30^{\\circ}$) located at\n$\\sim 8\\,\\rm{kPc}$, or a C\/O dwarf with high inclination angle\n($60^{\\circ} 0$ is the delay in the receiver's negative self-feedback. \nIn such system, \n${\\bf R}(t) = {\\bf S}(t+t_d)$ is a solution of the system, \nwhich can be easily verified by direct substitution \nin Eq.~\\ref{eq:voss}. AS has been observed in excitable\nmodels driven by white noise~\\cite{Ciszak03}, chaotic systems ~\\cite{Voss00,Pyragas08},\nas well as in experimental setups with semiconductor lasers~\\cite{Sivaprakasam01,Tang03} and electronic circuits~\\cite{Ciszak09}.\n\n\nAS has also been observed when the self-feedback was \nreplaced by parameter mismatches~\\cite{Corron05,Srinivasan12,Pyragiene13,Pyragiene15,Simonov14}, inhibitory dynamical loops~\\cite{Matias11,Matias14,Matias15,Pinto19}\nand noise at the receiver~\\cite{DallaPorta19}. It has been suggested that AS can emerge when the receiver dynamics is faster than the senders~\\cite{Hayashi16,Dima18,Pinto19,DallaPorta19}.\nFurthermore, unidirectionally coupled lasers reported both regimes: AS and the usual delayed\nsynchronization (DS, in which the sender predicts the activity of the receiver), depending on the difference between the transmission time and the feedback delay time~\\cite{Liu02,Tang03}. \nThe two regimes were observed to have the same stability of\nthe synchronization manifold in the presence of small perturbations due to noise or parameter\nmismatches~\\cite{Tang03}.\nNeuron models can also present a transition from positive to negative phase differences (from DS to AS) depending on coupling parameters~\\cite{Matias11,Matias14,Matias15,DallaPorta19}.\nTherefore, the study of anticipatory regimes in biological systems (not man-made) is \nreceiving more attention in the last years~\\cite{Hayashi2016Anticipatory,Stepp10,Washburn19,Roman19}.\n\n\nHere we employ spectral coherence and Granger causality (GC) measures to infer the direction of influence,\nas well as the phase difference between electrodes of the EEG from 11 subjects.\nWe verify, for all subjects, the existence of coherent activity in the alpha band ($f\\sim10$~Hz) between pairs of electrodes.\nWe also show that many of these pairs exhibit a unidirectional influence from one electrode to another and a phase difference that can be positive or negative.\nIn Sec.~\\ref{results} and ~\\ref{Appendix} we describe the experimental paradigm and EEG processing and analysis.\nIn Sec.~\\ref{results}, we report our results, showing that when we consider all the unidirectionally coupled pairs we verify that there is a diversity in the phase relation:\nthey exhibit in-phase, anti-phase, or out-of-phase synchronization with similar distribution of positive and negative phase differences (DS and AS, respectively).\nConcluding remarks and brief discussion of the significance of our findings for neuroscience are presented in Sec.~\\ref{conclusion}.\n\n\n\n\n\n\n\n\n\n\n\\section{\\label{results}Results}\n\n\nThe experiment consists in 400 trials of a GO\/NO-GO task.\nIn each trial a pair of\nstimuli were presented after a waiting window of $300$~ms, which is the important interval for our analysis (see the green arrow in Fig.~\\ref{fig:task}(b)).\nDepending on the combination of stimuli, participants should press a button or not.\nOscillatory main frequency, synchronized activity and directional influence were estimated by \nthe power, coherence, phase difference and Granger causality spectra\nas reported in Matias et al.~\\cite{Matias14} (see more details in Sec.~\\ref{Appendix}). \n\n\n\n\n\\begin{figure}[h]\n\\centering\n\\begin{minipage}{0.48\\linewidth}\n\\begin{flushleft}(a)%\n\\end{flushleft}%\n\\centering\n\\includegraphics[width=0.6\\columnwidth,clip]{1acabeca.eps}\n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n\\vspace{0.2cm}\n\\begin{flushleft}(b)%\n\\end{flushleft}%\n\\centering\n\\includegraphics[width=0.9\\columnwidth,clip]{1btask.eps}\n\\end{minipage}\n\\caption{\n{\\bf Experimental paradigm.} \n(a) 10\/20 System of EEG electrodes placement employed in the experiments.\n(b) GO\/NO-GO task based on three types of stimulus with images of animals (A), plants (P), and people (H$_+$).\nAfter a waiting window of 300~ms, two stimulus were presented for 100 milliseconds, with a 1000 ms inter-stimulus-interval.\nIf both stimulus are animals (AA) the participant should press a button as quickly as\npossible (see Sec.~\\ref{Appendix} for more details). Here we analyzed the 300~ms before the stimulus onset.\n}\n\\label{fig:task}\n\\end{figure}\n\n\n\n\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.9\\textwidth,clip]{2GC439DS3-1.eps}\n\\caption{\n{\\bf Unidirectional causality with positive phase-lag characterizes the delayed synchronization regime (DS).}\nPower, coherence, Granger causality and phase spectra between electrodes $F_7$ and $F_{P1}$ for volunteer 439.\nThe pair is synchronized with main frequency \n$f_{peak}=11.4$~Hz (given by the peak of the coherence, grey dashed lines). \nThe Granger causality peak around $f_{peak}$ reveals a directional influence from site $F_7$ to $F_{P1}$ and the phase difference \nat the main frequency $\\Delta\\Phi_{F_7-F_{P1}}(f_{peak})=0.1727$~rad shows that $F_7$ leads $F_{P1}$ (with an equivalent time delay $\\tau=2.4$~ms).\n}\n\\label{fig:GC-DS}\n\\end{figure*}\n\n\nSynchronization between electrodes $l$ and $k$ \ncan be characterized by a peak in the coherence spectrum $C_{lk}(f_{peak})$.\nThe phase difference $\\Delta\\Phi_{l-k}$ at the peak frequency \n$f_{peak}$ provides the time delay $\\tau_{lk}$ between the electrodes. \nThe direction of influence is given by the \nGranger causality spectrum. Whenever an electrode $l$ strongly and\nasymmetrically G-causes $k$, we\nrefer to $l$ as the sender (S) and to $k$ as the receiver (R) and the link between $l$ and $k$ is considered \na unidirectional coupling from $l$ to $k$ (S $\\to$ R). \nAfter determining which electrode is the sender and which one is the receiver we analyze the sign \nof $\\Delta\\Phi_{S-R}$ to determine the synchronized regime. Unless otherwise stated we analyze only the unidirectionally connected pairs.\n\n\n\n\\subsection*{Delayed synchronization (DS): unidirectional causality with positive phase-lag}\n\nTypically when a directional influence is verified from A to B, \na positive time delay is expected, indicating that A's activity temporally precedes that of B~\\cite{Gregoriou09,Sharott05}. \nThis positive time delay characterizes the intuitive regime called delayed synchronization (DS, or also retarded synchronization) in which the sender is also the leader~\\cite{Tang03}. \nIn neuronal models the time delay between A and B can reflect the characteristic time scale of the synapses between A and B but \ncan also be modulated by local properties of the receiver region B~\\cite{Matias14,DallaPorta19}. \n\n\nIn Fig.~\\ref{fig:GC-DS} we show an example of DS between the sites F$_7$ and F$_{P1}$ for volunteer 439. \nPower and coherence spectra present a peak at $f_{peak}=11.4$~Hz. \nAt this frequency, the activity of F$_7$ G-causes F$_{P1}$, but not the other way around. \nThe positive sign of the phase $\\Delta\\Phi_{F_7-F_{P1}}(f_{peak})=0.1727$~rad indicates that the sender electrode F$_7$ \nleads the receiver electrode F$_{P1}$ with a positive time delay $\\tau=2.4$~ms.\n\n\n\n\n\n\n\\subsection*{Anticipated synchronization (AS): unidirectional causality with negative phase-lag}\n\n\n\\begin{figure*}[!th]\n\\centering\n\\includegraphics[width=0.9\\textwidth,clip]{3GC439AS5-1.eps\n\\caption{\n{\\bf Unidirectional causality with negative phase-lag characterizes anticipated synchronization (AS). }\nPower, coherence, Granger causality and phase spectra between sites $F_Z$ and $F_{P1}$ for volunteer 439.\nThe electrodes are synchronized with main frequency $f_{peak}=10.8$~Hz (given by the peak of the coherence, grey dashed lines).\nThe Granger causality peak around $f_{peak}$ reveals a directional influence from site \n$F_Z$ to $F_{P1}$. $F_Z$ G-causes $F_{P1}$, but the negative phase difference \nat the main frequency $\\Delta\\Phi_{F_Z-F_{P1}}(f_{peak})=-0.1969$~rad (which is equivalent to a time delay $\\tau=-2.9$~ms) indicates that\n$F_{P1}$ leads $F_Z$ in time. \n} \\label{fig:GC-AS}\n\\end{figure*}\n\n\n\nDespite the fact that phase differences and coherence patterns, \nhave been employed to infer the direction of the information flux\n~\\cite{Marsden01,Williams02,Schnitzler05,Sauseng08,Gregoriou09,Korzeniewska03},\nour results imply that if we consider only the coherence and phase-lag \nwe could infer the wrong direction of influence between the involved pairs. \nSuch counter-intuitive regime exhibiting unidirectionally causality \nwith negative phase difference has first been reported in the brain as a mismatch\nbetween causality and the sign of the phase difference in local field potential \nof macaque monkeys during cognitive tasks~\\cite{Brovelli04,Salazar12}.\nAfterwards, it has been reported that the apparent paradox could be explained in the \nlight of anticipated synchronization ideas~\\cite{Matias14}.\nHere we show that human EEG signals can also present unidirectional influence \nwith negative phase-lag. As far as we know, this is the first evidence of AS in human EEG data.\n\nAn example of anticipated synchronization between EEG electrodes is shown in Fig.~\\ref{fig:GC-AS}. The sites $F_Z$ and $F_{P1}$\nexhibit a peak at alpha band in the power and coherence spectra for $f_{peak}=10.8$~Hz (Fig.~\\ref{fig:GC-AS}) for volunteer 439. \nThe Granger causality spectra presents a peak from \n$F_Z$ to $F_{P1}$ but not in the opposite direction, indicating that\n$F_Z$ G-causes $F_{P1}$ at $f_{peak}=10.8$~Hz.\nHowever, the negative sign of the angle $\\Delta\\Phi_{F_Z-F_{P1}}(f_{peak})=-0.1969$~rad indicates that the activity of F$_Z$ lags behind the activity of F$_{P1}$. \nThe time delay associated to $\\Delta\\Phi_{F_Z-F_{P1}}(f_{peak})$ is $\\tau=-2.9$~ms. \n\n\nIt is worth mentioning that for linear phase responses, which is the case for a simple monochromatic sinusoidal function, \nthe phase delay and the group delay (defined by the derivative of phase with respect to frequency) are identical. \nIn this case, both phase and group delays may be interpreted as the actual time delay between the signals.\nFor time series that are synchronized in a broad frequency band, the group delay could be useful to estimate \nthe time difference between the signals. Indeed, a negative group delay has been associated with anticipatory dynamics~\\cite{Voss16,Voss16Negative,Voss18}\nand it is comparable to the time difference obtained by the cross-correlation function~\\cite{Voss16}.\nHere, we verified that some AS pairs present both negative phase delay and negative group delay (as in the example shown in Fig.~\\ref{fig:GC-AS}). \nHowever, this is not the case for all AS pairs in the analyzed data. We have found all possible combinations for the signs of phase and group delays for both DS and AS. \nA further investigation of the relation between phase delay and group delay in brain signals is out of the scope of this paper and should be done elsewhere.\"\n\n\n\n\\subsection*{Zero-lag synchronization (ZL)}\n\nZero-lag (ZL) synchronization has been widely documented in experimental data since its first report in the cat visual cortex~\\cite{Gray89}. \nIt has been related to different cognitive functions such as perceptual integration and the execution of coordinated motor behaviours ~\\cite{Roelfsema97,Varela01,Fries05,Uhlhaas09}. \nDespite many models showing that bidirectional coupling between areas promotes zero-lag synchronization~\\cite{Vicente08,Gollo14}, \nit is also possible to have ZL between unidirectional connected populations~\\cite{Matias14,Matias15,DallaPorta19}. \nIn these systems, nonlinear properties of the receiver region can compesate characteristic synaptics delays and the two systems synchronize at zero phase.\n\n\nWe consider zero-lag whenever $\\arrowvert \\Delta\\Phi_{S-R}(f_{peak})\\arrowvert < 0.1$~rad. \nIn Fig.~\\ref{fig:GC-ZL} we show power, coherence, Granger causality and phase spectra between electrodes $F_3$ and $F_{P2}$ for volunteer 439.\nThese sites are synchronized with main frequency\n$f_{peak}=10.4$~Hz and $\\Delta\\Phi_{F_3-F_{P2}}(f_{peak})=-0.0164$~rad which provides $\\tau=-0.2$~ms.\n\n\\begin{figure*}[!th]\n\\centering\n\\includegraphics[width=0.9\\textwidth,clip]{4GC439ZL4-2.eps}\n\\caption{\n{\\bf Unidirectional causality with zero-lag synchronization (ZL, defined by $\\Delta\\Phi \\simeq 0$). }\nPower, coherence, Granger causality and phase spectra between electrodes $F_3$ and $F_{P2}$ for volunteer 439. Sites are synchronized with main frequency (given by the peak of the coherence, brown dashed lines) \n$f_{peak}=10.4$~Hz. The Granger causality peak around $f_{peak}$ indicates that site $F_3$ unidirectionally influences $F_{P2}$. The time delay between both is almost zero $\\tau=-0.2$~ms ($\\Delta\\Phi_{F_3-F_{P2}}(f_{peak})=-0.0164$~rad).\n}\n\\label{fig:GC-ZL}\n\\end{figure*}\n\n\n\\subsection*{Anti-phase synchronization}\n\nParticipants can also exhibit anti-phase synchronization between electrodes.\nWe define anti-phase synchronization (AP) when $ \\pi - 0.1 < \\arrowvert \\Delta\\Phi_{S-R}(f_{peak})\\arrowvert < \\pi + 0.1$~rad. \nIn Fig.~\\ref{fig:GC-AP} we show power, coherence, Granger causality and phase spectra between electrodes $O_2$ and $C_3$ for volunteer 439. The site $O_2$ G-causes $C_3$ \nand the time delay between them is $\\tau= 47.5$~ms which is almost half of a period for the $f_{peak}=10.4$~Hz.\n\n\n\\begin{figure*}[!th]\n \\centering\n \\includegraphics[width=0.9\\textwidth,clip]{5GC439AF19-9.eps}\n \\caption{\n {\\bf Unidirectional causality with anti-phase synchronization (AP, defined by $\\Delta\\Phi \\simeq \\pm\\pi$).}\nPower, coherence, Granger causality and phase spectra between electrodes $O_2$ and $C_3$ for volunteer 439.\nThe activity of the electrodes are synchronized with main frequency $f_{peak}=10.4$~Hz (grey dashed lines). \nThe Granger causality peak around $f_{peak}$ reveals a directional influence from $O_2$ to $C_3$ and \nthe phase spectrum shows that $\\Delta\\Phi_{O_2-C_3}(f_{peak})=3.1031$~rad (which provides $\\tau = 47.5$~ms).\n}\n \\label{fig:GC-AP}\n\\end{figure*}\n\n \n \n\\begin{table}[!ht]\n\\centering\n\\caption{\n{\\bf Number of unidirectionally connected pairs for all subjects together:} \nseparated by phase-synchronization regime along the lines and by the direction of influence along the columns.\n}\n\\begin{tabular}{|c| c| c| c| c |}\n\\hline\n& Unidirectional & Back-to-Front & Lateral & Front-to-Back \\\\ \\hline\n Total & 686 & 430 &90 & 166 \\\\ \\hline\nZL &93 & 39 &25 & 29 \\\\ \\hline\nDS(1) & 77 & 25 &14 & 38 \\\\ \\hline\nAS(1) & 99 & 51 &27 & 21 \\\\ \\hline\nAP & 174 & 135 &11 & 28 \\\\ \\hline\nDS(2) & 108 & 83 &4 & 21 \\\\ \\hline\nAS(2) & 135 & 97 &9 & 29 \\\\ \\hline\n\\end{tabular}\n\\label{tab:totalnumbers}\n \\end{table}\n\n\n \n \n\n\\subsection*{Phase relation diversity across pairs and subjects}\n\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.99\\columnwidth,clip]{6circjuntos.eps}\n \\caption{\n {\\bf Circular phase differences distribution.} \n The pairs are separated into six groups relative to their phase-synchronization regime: \n zero-lag (ZL, dark gray), anti-phase (AP, light gray), delayed synchronization in the first quadrant (DS(1),dark blue),\n delayed synchronization in the second quadrant (DS(2), light blue),\n anticipated synchronization in the fourth quadrant (AS(1), dark red), \n anticipated synchronization in the third quadrant (AS(2), light red). \n (a) Phase of all 686 unidirectionally connected pairs: (b) 430 pairs showing back-to-front influence, (c) 90 pairs within lateral flux, (d) 166 pairs presenting front-to-back influence.\n }\n \\label{fig:all_flux}\n\\end{figure}\n\n\nReliable phase relation diversity is a general property of brain oscillations.\nIt has been reported on\nmultiple spatial scales, ranging from very small spatial scale (inter-electrode distance $<900$ mm) in macaque~\\cite{Maris13,Dotson14},\nto a large spatial scale (using magnetoencephalography) in humans~\\cite{Van15}.\nHowever, the functional significance of phase relations in neuronal signals is not well defined. \nIt has been hypothesized that it may support effective neuronal communication by enhancing neuronal selectivity and promoting segregation of multiple information streams~\\cite{Maris16}.\n\n\nConsidering the 19 electrodes per subject, the number of analyzed pairs is 171 for each volunteer which corresponds to 1881 pairs in total. \nAmong these pairs, 1394 presented a peak in the coherence spectrum at the alpha band. Regarding the Granger causality spectra, 686 pairs presented an unidirectional \ninfluence and 358 a bidirectional influence. \nIn Fig.~\\ref{fig:all_flux}(a) we show the phase-difference distribution of all 686 unidirectionally connected pairs for all volunteers in a circular plot. \nIn Figs.~\\ref{fig:all_flux}(b),(c),(d) we show all the pairs separated by the direction of influence: from the back to the front (430), \nlateral flux (90) and from the front to the back (166), respectively. The colors represent the four different synchronized regimes mentioned before: \nDS (blue for positive phase: $0.1 < \\Delta\\Phi_{S-R}(f_{peak}) < \\pi - 0.1$~rad), AS (red for negative phase: $-\\pi+0.1 < \\Delta\\Phi_{S-R}(f_{peak}) < -0.1$~rad), \nZL (dark grey for close to zero-phase: $\\arrowvert \\Delta\\Phi_{S-R}(f_{peak})\\arrowvert < 0.1$~rad) \nand AP (light grey for phase close to $\\pm\\pi$: $\\pi - 0.1 < \\arrowvert \\Delta\\Phi_{S-R}(f_{peak})\\arrowvert < \\pi + 0.1$~rad). \nWe have also separated the DS and AS regimes into two different subcategories: DS(1) for phase in the first quadrant (dark blue), DS(2) for phase in the second quadrant (light blue), \nAS(1) for phase in the fourth quadrant (dark red) and AS(2) for phase in the third quadrant (light red). \nThe number of pairs in each situation are shown in Table~\\ref{tab:totalnumbers} and in Fig.~\\ref{fig:hist}.\n \n \n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.8\\columnwidth,clip]{7histogram.eps}\n \\caption{\n {\\bf Histograms for number of pairs in each synchronized regime.}\n The colors indicate phase-synchronization regime. (a) Electrode pairs are separated by direction of influence: all unidirectional pairs, back-to-front influence, front-to-back and lateral direction. \n (b) All unidirectional pairs separated per volunteer. \n}\n \\label{fig:hist}\n\\end{figure}\n\n\n \n \nThe total number of synchronized and unidirectionally connected pairs varies among volunteers, \nas well as the distribution of phases. All subjects present DS, AS, ZL and AP pairs (see Fig.~\\ref{fig:hist}(b)). \nHowever, one subject does not present AS(1). All subjects present back-to-front, lateral and front-to-back influence \nand more pairs with back-to-front than front-to-back direction of influence. \nConsidering only the back-to-front pairs, there are more AP than ZL synchronized regimes. \nThis is also true if we compare all pairs in the second and third quadrant (AP, DS(2) and AS(2)) with the ones in the first and fourth (ZL, DS(1), AS(1)).\n\nAs illustrative examples, in Fig.~\\ref{fig:circjun} we show the direction of influence between some pairs \nthat have the same unidirectional back-to-front Granger for at least 4 subjects and their respective phases. \nAlmost all pairs that have the electrodes $P_Z$, $P_3$ and $P_4$ as the sender present phases close to anti-phase (AP, DS(2), AS(2)), \nwhereas almost all the pairs in which the sender is $F_Z$, $T_3$ or $T_4$ are synchronized close to zero-lag (ZL, DS(1), AS(1)).\n\nRegarding back-to-front influences, no pair presented the same Granger causal relation for 9 or more subjects. \nThree pairs exhibited same unidirectional relation for 8 volunteers: $P_Z \\to F_{7}$, $P_3 \\to F_{P2}$, $O_1 \\to F_{4}$; \nother 3 pairs presented the same unidirectional relation for 7 subjects: $P_3 \\to F_{4}$, $P_3 \\to F_{8}$, $O_1 \\to F_{P2}$.\nTen pairs had same Granger causal relation for 6 volunteers: $F_Z \\to F_{P1}$, $P_3 \\to F_{P1}$, $P_3 \\to F_{3}$, $C_Z \\to F_{8}$, $C_Z \\to T_{3}$, $C_4 \\to F_{P1}$, $C_4 \\to F_{7}$, $C_4 \\to F_{3}$, $O_1 \\to F_{P1}$, $O_2 \\to F_{4}$. \nAll these 16 pairs had none or only one other subject presenting the opposite direction of the Granger causality. \nOut of these 16 pairs, only $F_Z \\to F_{P1}$ is mostly synchronized close do ZL as shown in Figs.~\\ref{fig:all_flux}(a) and (b), \nall others are mostly synchronized close to AP as in Figs.~\\ref{fig:all_flux}(c)-(f).\n\n\n\n\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.99\\columnwidth,clip]{8juntos.eps}\n \\caption{\n {\\bf Illustrative examples of unidirectionally connected pairs and their phase relations.} \n (a) and (b) Example of pairs with the majority of phase differences in the first and the fourth quadrants (ZL, DS(1), AS(1)): \n $F_Z\\rightarrow F_{P1}$, $F_Z \\to F_{P2}$, $T_3 \\rightarrow F_{P1}$ and $T_4 \\rightarrow F_{P2}$. \n (c) to (f) Example of pairs with the majority of phase differences in the second and the third quadrants \n (AP,DS(2),AS(2)). $P_Z$, $P_{3}$ and $P_{4}$ are well connected senders. All the chosen pairs are synchronized with same direction of influence for at least 4 subjects.\n } \n \\label{fig:circjun}\n\\end{figure}\n\n\n\\section{\\label{conclusion}Conclusion}\n\n\nWe show that human EEG can simultaneously present unidirectional causality and diverse phase relations between electrodes.\nOur findings suggest that the human brain can\noperate in a dynamical regime where the information flow and relative phase-lag have opposite signs.\nTo the best of our knowledge this is the first evidence of unidirectional influence accompanied by negative phase differences in EEG data. \nThis counter-intuitive phenomena have been previously reported as anticipated synchronization in monkey LFP~\\cite{Matias14,Brovelli04,Salazar12},\nin neuronal models~\\cite{Ciszak03,Matias11,Pyragiene13,Sausedo14,Simonov14} and in physical systems~\\cite{Sivaprakasam01,Tang03,Corron05,Ciszak09,Srinivasan12}.\nTherefore, we propose that this is the first verification of anticipated synchronization in EEG signals and in human brains.\n\nStudies estimating the actual brain connectivity using data from EEG signals should consider many relevant issues such as~\\cite{Brette12book}:\nthe importance of common reference in EEG to estimate phase differences~\\cite{Thatcher12}\nand the effects of volume conduction for source localization~\\cite{Nunez97,Van98}.\nOur findings suggest that it is also important to take into account the possible existence of AS in connectivity studies \nand separately analyze causality and phase relations. It is worth mentioning that, it has been shown that for enough data points the \nGranger causality is able to distinguish AS and DS regimes~\\cite{Hahs11}. \nHowever, for very well behaved time series the reconstruction of the connectivity can be confused by the phase~\\cite{Vakorin14}.\n\n\nOur results open important avenues for investigating how neural oscillations contribute to the neural \nimplementation of cognition and behavior as well as for studying the functional significance of phase diversity~\\cite{Maris13,Maris16}. \nFuture works could investigate the relation between anticipated synchronzation in brain signals and anticipatory behaviors~\\cite{Stepp10} such as anticipation in human-machine interaction~\\cite{Washburn19}\nand during synchronized rhythmic action~\\cite{Roman19}.\nIt is also possible to explore the relation between consistent phase differences and behavioral data such as learning rate, reaction time and task performance during different cognitive tasks .\nNeuronal models have shown that spike-timing dependent plasticity and the DS-AS transition together \ncould determine the phase differences between cortical-like populations~\\cite{Matias15}. \nHowever, an experimental evidence for the relation between learning and negative phase differences is still lacking. \n\n\nWe also suggest that our study can be potentially interesting to future researches on \nthe relation between inhibitory coupling, oscillations and communication between brain areas.\nOn one hand, inhibition is considered to play an important role to establish the oscillatory alpha activity, \nin particular, allowing selective information processes~\\cite{Klimesch07}. \nOn the other hand, according to the anticipated synchronization in neuronal populations model presented in Ref.~\\cite{Matias14},\na modification of the inhibitory synaptic conductance at the receiver population can modulate the phase relation between sender and receiver, \neventually promoting a transition from DS to AS. Therefore, we suggest that the inhibition at the receiver region can control the phase difference between cortical areas, which\nhas been hypothesized to control the efficiency of the information exchange between these areas,\nvia communication through coherence~\\cite{Fries05,Bastos15}.\n\n\n\\section{\\label{Appendix}Appendix: Methods}\n\n\n\\subsubsection*{Subjects}\nWe analyzed data from 11 volunteers \n(10 women, 1 man, all right-handed) who signed to indicate informed consent to participate in the experiment.\nThe youngest was 32 years old and the oldest 55 years old (average 45.7 and standard deviation 7.8).\nAll subjects were evaluated by both psychiatrist and psychologist.\nExclusion criteria\nwere: perinatal problems, cranial injuries with loss of consciousness and neurological deficit,\nhistory of seizures, medication or other drugs 24 hours before the recording, presence of psychotic\nsymptoms in 6 months prior the study and the presence of systemic and neurological diseases.\nThe experiment was not specifically designed to investigate \nthe phenomena of anticipated synchronization in humans and the data analyzed here were first analyzed in Ref.~\\cite{AguilarDomingo13}.\nThe entire experimental protocol was approved by the Commission of Bioethics of the University of Murcia (UMU, project: Subtipos electrofisiol\u00f3gicos y mediante estimulaci\u00f3n el\u00e9ctrica transcraneal del Trastorno por D\u00e9ficit de Atenci\u00f3n con o sin Hiperactividad).\n\n\\subsubsection*{EEG recording}\n\nThe electroencephalographic data recordings were carried out at the Spanish\nFoundation for Neurometrics Development (Murcia, Spain) center using a Mitsar 201M amplifier\n(Mitsar Ltd), a system of 19 channels with auricular reference.\nData were digitized at a frequency of 250 Hz. \nThe electrodes were positioned according to the international 10-20 system using\nconductive paste (ECI ELECTRO-GEL). Electrode impedance was kept $<5$~K$\\Omega$. \nThe montage (Fig.~\\ref{fig:task}(a)) include three midline sites (F$_Z$, C$_Z$ and P$_Z$) and eight sites over each hemisphere\n(F$_{P1}$\/F$_{P2}$,F$_{7}$\/F$_{8}$,F$_{3}$\/F$_{4}$,T$_{3}$\/T$_{4}$,C$_{3}$\/C$_{4}$,P$_{3}$\/P$_{4}$,T$_{5}$\/T$_{6}$ and O$_{1}$\/O$_{2}$).\nThe acquisition was realized by WinEEG software (Version 2.92.56).\nEEG epochs with excessive amplitude ($>50$~$\\mu$V) were automatically\ndeleted. Finally, the EEG was analyzed by a specialist in neurophysiology to reject epochs with\nartifacts. \n\n\n\n\\subsubsection*{Experimental task}\n\nThe EEG data were recorded while subjects performed a GO\/NO-GO task (also called visual continuous performance task, VCPT).\nParticipants sat in an ergonomic chair 1.5 meters away\nfrom a $17''$ plasma screen. Psytask software (Mitsar Systems) was used to present the images.\nThe VCPT consists of three types of stimuli: twenty images of animals (A), twenty images\nof plants (P), twenty images of people of different professions (H$_+$). \nWhenever H$_+$ was presented, a $20$~ms-long artificial sound tone frequency\nwas simultaneously produced. The tone frequencies range from 500 to $2500$~Hz, in intervals of $500$~Hz.\nAll stimuli were of equal size and brightness. \n\nIn each trial a pair of\nstimuli were presented after a waiting window of $300$~ms, which is the important interval for our analysis (see the green arrow in Fig.~\\ref{fig:task}(b)).\nEach stimulus remains on the screen for $100$~ms, with a $1000$~ms inter-stimulus-interval.\nFour different kinds of pairs of stimuli were employed: AA, AP, PP and PH$_+$. \nThe entire experiment consists in 400 trials (the four kinds of pairs were randomly distributed and each one appeared 100 times).\nThe continuous set occurs when A is presented as the first stimulus, so the subject\nneeded to prepare to respond. An AA pair corresponds to a GO task and the participants are supposed to press a button as quickly as\npossible. An AP pair corresponds to a \nNO-GO task and the participants should\nsuppress the action of pressing the button. \nThe discontinuous set, in which P\nis first presented, indicates that one should not respond (independently of the second stimuli). IGNORE task occurred with PP\npairs and NOVEL when PH$_+$ pairs appeared.\nParticipants were trained for about five minutes before beginning the experimental trials. They rested for\na few minutes when they reached the halfway point of the task. The experimental session lasted $\\sim30$~min.\n\n\n\n\n\\subsubsection*{EEG processing and analysis}\n\nThe Power, Coherence, Granger causality and phase difference spectra\nwere calculated following the methodology reported\nin Matias et al.~\\cite{Matias14} using the auto-regressive modeling method (MVAR) implemented in the MVGC Matlab\ntoolbox~\\cite{Barnett14}. Data were\nacquired while participants were performing the GO\/NO-GO visual pattern\ndiscrimination described before. \nOur analysis focuses on $30000$ points \nrepresenting the waiting window of $400$ trials ending with the\nvisual stimulus onset (green arrow in Fig.~\\ref{fig:task}(b)).\nThis means that in each trial, the 300-ms pre-stimulus interval consists of \n75 points with a 250-Hz sample rate.\n\nThe preprocess of the multi-trial EEG time series consists in\ndetrending, demeaning and normalization of each trial. \nRespectively, it means to subtract from the time series the best-fitting line, \nthe ensemble mean and divide it by the temporal standard deviation.\nAfter these processes each single trial can be considered \nas produced from a zero-mean stochastic process.\nIn order to determine an optimal order for the MVAR model we obtained\nthe minimum of the Akaike Information Criterion (AIC)~\\cite{Akaike74} \nas a function of model order. The AIC dropped\nmonotonically with increasing model order up to 30. \n\n\nFor each pair of sites $(l,k)$ we\ncalculated the spectral matrix element\n$S_{lk}(f)$~\\cite{Brovelli04,Lutkepohl93}, from which the coherence\nspectrum $C_{lk}(f) = |S_{lk}|^2\/[S_{ll}(f)S_{kk}(f)]$ and the phase\nspectrum $\\Delta\\Phi_{l-k}(f) =\n\\tan^{-1}[\\mbox{Im}(S_{lk})\/\\mbox{Re}(S_{lk})]$ were calculated. \nA peak of $C_{lk}(f)$ indicates synchronized oscillatory activity at the\npeak frequency $f_{peak}$, with a time delay $\\tau_{lk} =\n\\Delta\\Phi_{lk}(f_{peak})\/(2\\pi f_{peak})$. We only consider $7