Quantum state inputs for tailored and externally corrected coupled cluster methods

Abstract

Methods, systems, and apparatus for performing electronic structure calculations. In one aspect, a method includes preparing, by quantum computation, approximations of a ground state in an active space of a quantum system; measuring, by quantum computation, the approximations of the ground state to obtain a plurality of cluster amplitudes; and processing, by classical computation, the plurality of cluster amplitudes to simulate the quantum system, comprising providing the plurality of cluster amplitudes as input to a coupled cluster method that recovers dynamic correlation in the quantum system.

Description

[0001]Attorney Docket No. 56113-0503WO1 QUANTUM STATE INPUTS FOR TAILORED AND EXTERNALLY CORRECTED COUPLED CLUSTER METHODS BACKGROUND This specification relates to quantum computing and quantum simulation. Efficient simulation of chemistry and materials requires a robust protocol to handle two types of quantum correlations. One type of quantum correlation results from some systems requiring multiple reference states and is referred to as multi-reference or static correlation. Another type of quantum correlation results from some systems having states with small amplitudes that can contribute to the energy or an observable's expectation in a substantial way, e.g., additive over an exponentially large Hilbert space, and is referred to as dynamic correlation. Existing quantum electronic structure methods can provide robust strategies for handling the static correlation problem but have no efficient route to the dynamic correlation component. Some classical electronic structure methods use coupled-cluster (CC) theory to recover dynamic correlation. In CC theory, higher-rank excitations are systematically included in the cluster operator in order to establish a hierarchy of increasingly accurate approximations that converge toward the full configuration interaction limit. The theory is based on an exponential ansatz acting on a reference wave function, e.g., a Slater determinant obtained from Hartree–Fock (HF), density functional theory, or other independent particle models, which is assumed to provide a reasonable zero-th order description of the correlated ground-state wave function. The CC wave function is determined by so-called cluster amplitudes obtained by solving nonlinear energy-independent CC equations. To make CC approximations numerically feasible, the cluster operator can be defined by low-rank excitations, e.g., single and double excitations. Some quantum algorithms recover dynamic correlation through measurement of reduced density matrices (RDMs) and low-order perturbation theory. However, these methods can have considerable measurement overheads. SUMMARY This disclosure describes techniques to recover correlation using quantum state inputs for coupled-cluster methods. In general, one innovative aspect of the subject matter described in this specification can be implemented in a method for performing an electronic structure Attorney Docket No. 56113-0503WO1 calculation. The method includes preparing, by quantum computation, approximations of a ground state in an active space of a quantum system; measuring, by quantum computation, the approximations of the ground state to obtain a plurality of cluster amplitudes; and processing, by classical computation, the plurality of cluster amplitudes to simulate the quantum system, comprising providing the plurality of cluster amplitudes as input to a coupled cluster method that recovers dynamic correlation in the quantum system. Other implementations of these aspects includes corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more classical and quantum computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions. The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations processing the plurality of cluster amplitudes to simulate the quantum system comprises solving coupled cluster equations to calculate an energy or set of observables for the quantum system. In some implementations solving coupled cluster equations to calculate an energy or set of observables for the quantum system comprises: fixing values of the plurality of cluster amplitudes in the coupled cluster equations to the measured plurality of cluster amplitudes; solving for remaining cluster amplitudes in the coupled cluster equations to obtain a coupled cluster wavefunction; and calculating the energy or set of observables for the quantum system using the coupled cluster wavefunction. In some implementations measuring the approximations of the ground state to obtain a plurality of cluster amplitudes comprises performing a polynomial number of measurements to a predetermined fixed error. In some implementations the plurality of cluster amplitudes comprise computational basis amplitudes. In some implementations the plurality of cluster amplitudes comprise configuration interaction coefficients in the active space. Attorney Docket No. 56113-0503WO1 In some implementations preparing the approximations of the ground state in the active space of the quantum system comprises performing phase estimation or imaginary time evolution. In some implementations measuring the approximations of the ground state to obtain the plurality of cluster amplitudes comprises performing state tomography, shadow tomography, or sign tomography with sampling. In some implementations the coupled cluster method comprises a tailored coupled cluster method. In some implementations the cluster amplitudes comprise cluster amplitudes up to second order in active space. In some implementations the coupled cluster method comprises an externally- corrected coupled cluster method. In some implementations the cluster amplitudes comprise cluster amplitudes up to fourth order. In some implementations the quantum system comprises a quantum system with electronic structure, for example a chemical or material. In some implementations the quantum hardware comprises a noisy intermediate scale quantum computer. In some implementations the quantum hardware comprises a fault tolerant quantum computer. The subject matter described in this specification can be implemented in particular embodiments so as to realize one or more of the following advantages. Examples of the presently described techniques can improve on known methods in several ways. For example, many known approaches for determining correlation effects in an electronic structure computation require an accurate determination of reduced density matrices, which is resource-intensive with respect to the required number of state preparation and measurement repetitions. However, examples of the presently described techniques does not rely on the determination of reduced density matrices. Instead, the techniques integrate classical electronic structure methodologies and quantum computation to lower the measurement requirements – only a polynomial number of measurements for fixed precision. This reduces the computational resource requirements and also enables larger quantum systems to be treated. In addition, examples of the presently described techniques can achieve improved resilience to device noise through error mitigation and error correction techniques. Attorney Docket No. 56113-0503WO1 A system implementing the presently described classical-quantum techniques can achieve technical improvements on both the classical computing and quantum computing side. For example, using a quantum computer to compute cluster amplitudes in active space requires The details of one or more implementations of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. BRIEF DESCRIPTION OF THE DRAWINGS FIG.1 is a block diagram of an example system for capturing electron correlation in an electronic structure computation using classical and quantum computation. FIG.2 is a flow diagram of an example process for performing an electronic structure calculation using classical and quantum computation. FIG.3 depicts an example quantum computing system. FIG.4 illustrates a schematic diagram of an exemplary classical processor system. DESCRIPTION This specification describes techniques for determining electron correlation in an electronic structure computation using hybrid classical-quantum computation. A quantum state that approximates the ground state in an active space representation of a target electronic structure system is determined. A quantum computation is performed to prepare the quantum state and a polynomial number of wavefunction amplitudes, e.g., computational basis amplitudes, are measured from the quantum state. These amplitudes are provided as input into a coupled cluster method, e.g., a tailored or externally corrected coupled cluster method. FIG.1 is a block diagram of an example system for capturing dynamic correlation in an electronic structure computation using classical and quantum computation. The example system 100 includes a classical processor 102 and a quantum processor 104. The classical processor 102 and quantum processor 104 can electronic communications over one or more networks, or can exchange communications in another way, such as over one or more wired or wireless connections. The classical processor 102 is configured to perform classical computations. The quantum processor 104 is configured to perform quantum computations. For convenience, Attorney Docket No. 56113-0503WO1 the classical processor 102 and quantum processor 104 are illustrated as separate entities. For example, the quantum processor 104 can be a quantum processor that is operated by an external third party. However, in some implementations the classical processor 102 can be included in the quantum processor 104. That is, the quantum processor 104 can also include components for performing classical computing operations. Generally, the classical computing components of the classical processor can be implemented as one or more classical computers having physical hardware like that described with respect to FIG.4 and the quantum computing components of the quantum processor 104 can be implemented as quantum computing devices having physical hardware like that described with respect to FIG.3. Example system 100 is configured to perform operations to capture electron correlation in an electronic structure computation using classical and quantum computation. The classical processor 102 is configured to receive input data specifying the simulation problem 108. For example, the classical processor 102 can receive data specifying a target quantum system to be simulated, e.g., a chemical or material. In some implementations the data 108 can also specify an active space for the quantum system, i.e., denote a subset of the system’s orbitals and electrons as active atomic orbitals. In other implementations the classical processor 102 can be configured to select an active space for the quantum system. In some implementations the data 108 can also specify a coupled cluster method to use to simulate the quantum system. The selection of an active space and coupled cluster method can depend on various factors, e.g., the quantum system being simulated and whether the quantum processor 104 is a NISQ or fault tolerant device. The classical processor 102 is configured to determine 110 cluster amplitudes (see, e.g., Equations (2) and (9) below) for use in determining a cluster wavefunction (see, e.g., Equation (1) below). The classical processor 102 provides data 112 specifying the quantum system Hamiltonian and wavefunction as a quantum circuit to the quantum processor 104. The quantum processor 104 is configured to receive the data 112 and to prepare a qubit encoding of an approximation of the ground state of the quantum system in the active space (e.g., using the Jordan-Wigner transformation to map fermionic operators to qubit operators). For example, the quantum processor 104 can perform 114 a phase estimation algorithm or imaginary time evolution to prepare the approximate ground state of the quantum system in active space. The quantum processor 104 can then measure 114 an observable with respect to the prepared ground state to obtain measurement results 116 that correspond to the cluster amplitudes, e.g., which can be efficiently converted to configuration interaction (CI) Attorney Docket No. 56113-0503WO1 coefficients. For example, the quantum processor 104 can be configured to repeatedly: prepare the (approximate) ground state in active space using phase estimation or imaginary time evolution techniques, apply a quantum circuit that evolves the (approximate) ground state according to the observable, and repeatedly measure the evolved quantum state to obtain the cluster amplitudes. An example measurement strategy is described below with reference to FIG.2. The quantum processor 104 is configured to transmit data representing results of the measurement operations 116 to the classical processor 102. The classical processor 102 is configured to receive the transmitted data 116 and perform 118 the coupled cluster method using the cluster amplitudes in active space as input. Example coupled cluster methods are described below. The classical processor 102 is configured to process a quantum measurement output by the coupled cluster method to solve the electron correlation problem, e.g., to compute an energy or set of observables for the quantum system. The classical processor 102 is configured to output data representing the solution to the electron correlation problem 120. Coupled cluster methods determine the correlated electronic structure of a quantum system through the construction of an approximation of an exact many-body wavefunction of the quantum system that incorporates electron correlation effects. The exact many-body wavefunction is approximated using an exponential ansatz |^^^^^ ൌ ^^ ^் |^^^^, (1) where |^^^^^ is the approximation of the exact many-body wavefunction, ^^^ is the clusteroperator, and |^^^^ is a Hartree-Fock (or Fermi vacuum) reference determinant. The clusteroperator , ^^^ is formed by linear combination of individual excitation operators ^^^௩ overexcitation levels ^^, that is ^^^ ൌ ^^^^ ^ ^^^ଶ ^ ^^^ଷ ^ ⋯ where ^^^^ represents the single excitations,^^^ଶrepresents the double excitations, etc. Truncating the linear combination at a certain excitation level ^^^^௫yields a hierarchy of truncated CC methods: CC singles (CCS, ^^^^௫= 1), CC singles and doubles (CCSD, ^^^^௫= 2), CC singles, doubles, and triples (CCSDT, ^^^^௫= 3), etc. Each excitation operator can be generally defined as ^^^௩ ൌ^∑^^^…^^^^^^^^……^^^ற^^^^ற ^ ^^^ற^ …^^^^^^^^^^^^…, (2) Attorney Docket No. 56113-0503WO1 where ^^^^^^^^……are cluster amplitudes for excitation level ^^, ^^^,^^^றrepresent fermionic annihilation and creation operators, the indices ^^, ^^, ^^, . . . represent virtual orbital indices, ^^, ^^, ^^, . . . represent occupied orbitals in |^^^^. The cluster amplitudes aredetermined by solving a system of nonlinear equations derived from the Schrödinger equation. For example, the cluster amplitudes for excitation operator ^^^௩are solved for by projecting excited determinant manifolds |^^௩^ onto a similarity-transformed Hamiltonian: ^^^^ ≡ ^^^௩|^ഥ^|^^^^ ് 0 ^3^where ^^ is referred to as a residua ഥ ି ^் ^்௩ l vector, ^^ ≡ ^^ ^^^^ is the similarity-transformedHamiltonian, ^^ is the Hamiltonian characterizing the quantum system, |^^^^ is the reference determinant (a single Slater determinant composed of the lowest-energy (occupied) orbitals), and |^^௩^ is the excited determinant manifold generated by the cluster operator ^^^௩acting on the reference determinant. The resulting non-linear cluster amplitude equations can be solved iteratively. Due to the similarity transformation, the resulting Hamiltonian may not be Hermitian. By projecting with |^^^^ from the left, the single reference coupled cluster energy is obtained as ^^^^ ൌ ^^^^|^ഥ^|^^^^ ൌ ^^ுி ^ ^^^^^^^^ 1 ^ ^^^^^^^^^^1 ^^^^^||^^^^^ ^^^^^^^^^^^^^||^^^^^ (4) with the reference Hartree-Fock energy ^^ , the Fock mat^ுிrix elements ^^^, and the anti- symmetrized two-electron repulsion integrals in notation ^^^^^||^^^^^. Only ^^^^ and ^^^ଶamplitudes enter the single reference coupled cluster energy expression directly, independent of the truncation level, since the higher excitation cluster operators cannot produce fully contracted terms with the Hamiltonian. The implicit energy contribution of higher-order ^^^௩amplitudes originates from the coupling of all amplitudes through the projection equations. For singles and doubles (CCSD), the amplitude equations require projections of the singly and doubly excited determinants, i.e., 0ൌ ^^^^^|^ഥ^|^^^^ Attorney Docket No. 56113-0503WO1 (5) where Φ୧ୟis a slater determinant formed by removing one electron from the occupied orbital ^^ and placing it in the virtual orbital ^^, and Φ୧ୟ୨ୠis a slater determinant formed by removing from the occupied orbitals ^^ and j and placing them in the virtual orbitals ^^ and b. The tailored coupled cluster (TCC) ansatz aims to encode static correlation effects from an active space (AS) method in a single reference coupled cluster (SRCC) wavefunction through a split-amplitude ansatz ^^^ ൌ ^^^^௫௧ ^ ^^^^ௌ ^6^where ^^^^ௌrepresents the (AS) and ^^^^௫௧represents thecluster operator for the (remaining) external space. The amplitudes in the active space cluster operator ^^^^ௌcan be extracted from an exact or approximate active space wavefunction through the relationship of a linear configuration interaction (CI) ansatz and the exponential CC ansatz 1^ ^^^^ ^்௩ ൌ ^^ ^7^ where the CI excitation operators ^^^௩ are defined as the excitation operator ^^^௩, but with ^^^^^^^^……as corresponding amplitudes. A conversion from CI to CC amplitudes can be achieved by matching excitation levels and recursively determining the amplitudes, here up to four-fold excitations, as Evaluating these terms as Wick contractions yields the programmable expressions Attorney Docket No. 56113-0503WO1 (9) where index a set of virtual orbitals for the quantum system, ^^^^represents a single excitation amplitude from an occupied orbital ^^ to a virtual orbital ^^ and ^^^^^^represents a double excitation amplitude from a pair of occupied orbitals ^^^^ to a pair of virtual orbitals ^^^^. These expressions contain single “non-redundant” outer products of cluster operator amplitudes, e.g., ^^^^^^^^^^, and terms with permuted indices ensure the correct anti-symmetry of the cancelling exactly the corresponding pre-factor from the Taylor of the reference determinant in the underlying CI expansion is not equal to one, the above conversion equations can be normalized accordingly. TCC is generally applicable when the coefficient of the reference determinant extracted from the active space method is non-zero. With a set of CI amplitudes at hand, the corresponding spin-orbital cluster operator amplitudes can be built with the expressions above, yielding an active space cluster operator ^^^^ௌ. The orbital indices of the clusteramplitudes are fully contained in the active space, i.e., ^^^^^^ ... ^^^^^^ ...∈ ^^^^. Contrary to that,the external cluster amplitudes belonging to ^^^^௫௧comprise an orbital space where at least oneorbital index is not part of the active space, i.e., ^^^^^^ ... ^^^^^^ ...⊄ ^^^^. Then, the TCC energyfunctional is given by ^^^்^^^ ಲೄ்^^ ൌ ^^^^|൫^ഥ^ே^^ା ^்൯^|^^^^ ^10^ where ^ഥ^ே represents a normal-ordered Hamiltonian ^ഥ^ே ൌ ^ഥ^ െ ^^^^|^ഥ^|^^^^. Since theexternal and active space cluster operators commute by the active space energy can be obtained through the CC energy functional via Attorney Docket No. 56113-0503WO1 This are exact and can be viewed as optimized in presence of all higher-order cluster operator amplitudes through the active space method. For approximate active space methods, the mapping is not exact, and the resulting energy computed through the CC energy functional is not necessarily equivalent to the (variational) energy of the active space wavefunction. To retain the static correlation information in the TCC wavefunction, the active space amplitudes are kept frozen during optimization of the external amplitudes. That is, the following amplitude equations are solved: The algebraic expressions of the projection equations. Thus, the frozen active space cluster operator amplitudes impact the TCC solution through the active space energy, i.e., the energy evaluated directly from the frozen amplitudes, and additionally through the contractions with the external amplitudes. Once the active space cluster operator amplitudes are known and mapped to the corresponding orbitals in the full molecular orbital space, the standard SRCC framework is used to solve for the external amplitudes, with the only constraint that a certain stride of the full space amplitudes be frozen. In practice, this can be achieved by setting the stride of active space amplitudes in the CC residual vector to zero. As for standard singles and doubles CCSD, a perturbative triples correction, i.e., CCSD(T) can be employed, which is only evaluated from the external amplitudes for consistency. Externally corrected CC (ec-CC) builds a similar split-amplitude ansatz as TCC, but encodes static correlation into the SRCC wavefunction through higher-order cluster operators. Considering the un-truncated expressions for the singles and doubles residualequations only ^^^ଷ and ^^^ସ contribute to achieve the correct total excitation level, i.e., Attorney Docket No. 56113-0503WO1These for ^^^^ and ^^^ଶ,however, higher-order cluster operators cannot produce any further contributions to these terms. Hence, the traditional CCSD approach corresponds to the approximate case where ^^^ଷ= 0 and ^^^ସ = 0. This means that, adding the direct contributions of ^^^ଷ and ^^^ସ amplitudes(e.g., extracted from a high-quality correlated multi-reference wavefunction) to the ^^^^ and ^^^ଶprojection equations, an improved treatment of the overall electron correlation can be obtained, including non-dynamic correlation effects, compared to CCSD. Convergence tofull CI is guaranteed if the input ^^^ଷ and ^^^ସ amplitudes become exact. Thus, the ec-CCprocedure optimizes the ^^^^ and ^^^ଶ amplitudes in presence of (approximate) ^^^ଷ and ^^^ସ. Theresulting ec-CC cluster operator is given by ^^^ ൌ ^^^ ^^^௨௧^^^௨௧^^ ^^^ଶ ^ ^^^ଷ ^ ^^^ସ^14^ where the ^^^ଷ and ^^^ସ cluster amplitudes are extracted from some input wavefunction. If thewavefunction only covers a subset of the molecular orbitals, i.e., in case of an active spacemethod, the operator is modified to project the sliced ^^^ଷ and ^^^ସ amplitudes onto the fullorbital space, like in TCC, that is ^^^ ൌ ^^^^ ^ ^^^ଶ ^ ^^^ଷ^^^ ^ௌଷ ^ ^^^ସ^^^ସ^ௌ^15^with the appropriate projection operators ^^^ଷ and ^^^ସ. For ec-CC, the same recursion relationsto extract cluster operator amplitudes from a CI-like wavefunction apply, see Equation (8). If an active space wavefunction is used as input, the convergence guarantee toward full configuration interaction does not hold anymore, however, improvements can still be achieved by including the dominant T3 and T4 amplitudes from the multi-reference treatment compared to completely neglecting the triples Secretand quadruples. Attorney Docket No. 56113-0503WO1 FIG.2 is a flow diagram of an example process 200 for performing an electronic structure calculation using classical and quantum computation. For convenience, the process 200 will be described as being performed by a system of one or more classical and quantum computing devices located in one or more locations. For example, the system 100 shown in FIG.1, appropriately programmed, can perform example process 200. The system receives data representing an electron correlation problem (step 202). For example, the system can receive data specifying a quantum system with electronic structure, e.g., a chemical or material, and properties of the quantum system to simulated, e.g., an energy of the quantum system. The system can use the received data to determine or select cluster amplitudes, e.g., CI coefficients, in an active space representation of the quantum system (step 204). The system can provide instructions to the quantum computing device that instruct the quantum computing device to measure values of the cluster amplitudes in the active space representation. The system uses the quantum computing device to prepare an approximation of a ground state of the quantum system in the active space (step 206). For example, the quantum computing device can implement a phase estimation algorithm or perform imaginary time evolution to prepare the approximation of the ground state of the quantum system in the active space. Performing phase estimation introduces at least one source of error into example process 200 (statistical noise). Performing imaginary time evolution introduces at least two sources of error into example process 200. The system uses the quantum computing device to measure the prepared ground state and obtain values of the cluster amplitudes (step 208). The measurements performed by the system include measurements of overlaps of Slater determinants and the prepared ground state, see, e.g., Equation (5) above. The number of measurements performed at step 208 is polynomial to a predetermined fixed error. In some implementations the system can perform state tomography, shadow tomography, or sign tomography with sampling to measure the ground state. When a Jordan-Wigner mapping of fermions to qubits is used to map the ground state of the quantum system from a fermionic quantum state to a qubit state, e.g., a state of a register of qubits included in the quantum computing device, the Slater determinant overlaps correspond to overlaps with computational basis states. Since the electronic structure Hamiltonian can, without loss of generality, be chosen to be real when written in the computational basis, the eigenstates are real, so that overlaps of the quantum state prepared at step 206 with computational basis states are real. For example, the system can perform a Attorney Docket No. 56113-0503WO1 shadow protocol to perform the measurements. The system prepares a superposition of a reference state (e.g., a true vacuum |0^⊗n) and the state prepared at step 206 |ΨT^. This can be achieved by applying a quantum circuit that prepares |ΨT^ from the Hartree-Fock state |Φ0^ to a state that is a superposition of |Φ0^ and |0^⊗n. This state can be prepared using a single Hadamard gate and replacing the Pauli X gates needed to prepare |Φ0^ by their controlled versions (CNOT). Alternatively, a classical shadow protocol that randomizes only over number preserving (passive) Gaussian unitary operators U or a Clifford shadow protocol can be used. The system can then take computational basis state measurements of the superposition state, e.g., after random Gaussian rotations have been applied. The overlap with a computational basis state |φ^ is then computed as the expectation value of the non- Hermitian observables |0^ ^φ|, using that ^φ |ΨT^ = 2Tr[|0^^φ|ρ^ where |ρ^ is the superposition state. If this state preparation is noisy so that instead of the state ρ = |ρ^ ^ρ| the device prepares the state ρ’ = (1 − p) ρ + p ρnoise for some error probability p and density matrix ρnoise, then 2Tr[|0^^φ|ρ’^ = (1 − p) ^φ |ΨT^ + p^φ|ρnoise|0^. For several reasonable types of noise, including depolarizing noise, bit flip noise, and amplitude damping ^φ|ρnoise|0^ will be either zero or very small for all states |φ^ from a subspace of number of particles. This means that the overlaps can be approximated by quotients of overlaps computed from a shadow of the noisy state. For example, ^Φୟ|Ψ^^Tr^|0^^Φୟ| ᇱ^^^ ୟ ୧ ୧ ρ^^ൌ ^Φ ൌ^ ^ ^ ^^ where ^^^^is a single excitation amplitude from an occupied orbital ^^ to a virtual orbital ^^, Φ୧ୟis a slater determinant formed by removing one electron from the occupied orbital ^^ placing it in the virtual orbital ^^, Ψ^ୗis the ground state of the quantum system in the active space, Φ^is the reference determinant, Ψ^,is the state prepared at step 206, and ρ’ is the noisy superposition state. Even estimation from a finite shot shadow should therefore be resilient to device noise as long as (1−p) ^φ |ΨT^ is not too small. Contrary to this, without error mitigation techniques, expectation values such as that of, e.g., the electronic structure Hamiltonian, are usually first order sensitive to the noise strength p. In addition, a major concern in many quantum computing platforms are particle number dependent phases that can be caused, e.g., by background magnetic fields. These are often particularly problematic Attorney Docket No. 56113-0503WO1 for algorithms that prepare coherent superpositions of states of markedly different particle number such as |ρ^. However, since the input overlaps are all real and the relative signs of the coefficients are important, which are quotients of overlaps, correct and the computational basis state overlaps going into the enumerator all come from the same particle number sub- space, a particle number dependent phase results in a global phase affecting all coefficients, which can be corrected, e.g., by rotating the coefficients in the complex plane such that the largest coefficient or the principle component of all overlaps or is aligned with the real axis before either discarding the imaginary part or taking the absolute value multiplied with the sign of the real part of each coefficient. This renders the techniques used herein largely independent of uncontrolled particle number dependent phases and also implies some robustness against general phase errors. The system uses the classical computing device to process the set of cluster amplitudes to simulate the quantum system (step 210). For example, the system can provide the set of cluster amplitudes as input to a coupled cluster method that recovers dynamic correlation in the quantum system. For example, one coupled cluster method is tailored coupled cluster, where the cluster amplitudes include cluster amplitudes up to second order in active space (see, e.g., Equation (9) above). Measured values of the single and double excitations can be used to define coefficients ^^^^and ^^^^^^, which define respective cluster operators ^^^^and ^^ଶ^and are used to solve coupled cluster equations. Another coupled cluster method is externally-corrected coupled cluster, where the cluster amplitudes include cluster amplitudes up to fourth order in active space. Processing the plurality of cluster amplitudes to simulate the quantum system then includes solving corresponding coupled cluster equations (see, e.g., Equation (12) above), to calculate an energy or set of observables for the quantum system, e.g., by fixing values of the cluster amplitudes in the coupled cluster equations to the measured cluster amplitudes, solving for the remaining cluster amplitudes in the coupled cluster equations to obtain a coupled cluster wavefunction, and calculating the energy or set of observables for the quantum system using the coupled cluster wavefunction. Implementations of the subject matter and operations described in this specification can be implemented in digital electronic circuitry, analog electronic circuitry, suitable quantum circuitry or, more generally, quantum computational systems, in tangibly-embodied software or firmware, in computer hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. The Attorney Docket No. 56113-0503WO1 term “quantum computational systems” may include, but is not limited to, quantum computers, quantum information processing systems, quantum cryptography systems, or quantum simulators. Implementations of the subject matter described in this specification can be implemented as one or more computer programs, i.e., one or more modules of computer program instructions encoded on a tangible non-transitory storage medium for execution by, or to control the operation of, data processing apparatus. The computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially- generated propagated signal that is capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus. The terms quantum information and quantum data refer to information or data that is carried by, held or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two- level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states are possible. The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The Attorney Docket No. 56113-0503WO1 apparatus can optionally include, in addition to hardware, code that creates an execution environment for digital and / or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A digital computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and translated into a suitable quantum programming language, or can be written in a quantum programming language, e.g., QCL or Quipper. A computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program in question, or in multiple coordinated files, e.g., files that store one or more modules, sub- programs, or portions of code. A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g. qubits. Generally, a digital data communication network cannot transmit quantum data, however a quantum data communication network may transmit both quantum data and digital data. The processes and logic flows described in this specification can be performed by one or more programmable computers, operating with one or more processors, as appropriate, executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers. Attorney Docket No. 56113-0503WO1 For a system of one or more computers to be “configured to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by data processing apparatus, cause the apparatus to perform the operations or actions. For example, a quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions. Computers suitable for the execution of a computer program can be based on general or special purpose processors, or any other kind of central processing unit. Generally, a central processing unit will receive instructions and data from a read-only memory, a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof . The elements of a computer include a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital, analog, and / or quantum data. The central processing unit and the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a computer need not have such devices. Quantum circuit elements (also referred to as quantum computing circuit elements) include circuit elements for performing quantum processing operations. That is, the quantum circuit elements are configured to make use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data in a non-deterministic manner. Certain quantum circuit elements, such as qubits, can be configured to represent and operate on information in more than one state simultaneously. Examples of superconducting quantum circuit elements include circuit elements such as quantum LC oscillators, qubits (e.g., flux qubits, phase qubits, or charge qubits), and superconducting quantum interference devices (SQUIDs) (e.g., RF-SQUID or DC-SQUID), among others. In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively carry out instructions of a computer program by performing basic arithmetical, logical, and / or input / output operations Attorney Docket No. 56113-0503WO1 on data, in which the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to transmit data to and / or receive data from the quantum circuit elements through electrical or electromagnetic connections. Examples of classical circuit elements include circuit elements based on CMOS circuitry, rapid single flux quantum (RSFQ) devices, reciprocal quantum logic (RQL) devices and ERSFQ devices, which are an energy-efficient version of RSFQ that does not use bias resistors. In certain cases, some or all of the quantum and / or classical circuit elements may be implemented using, e.g., superconducting quantum and / or classical circuit elements. Fabrication of the superconducting circuit elements can entail the deposition of one or more materials, such as superconductors, dielectrics and / or metals. Depending on the selected material, these materials can be deposited using deposition processes such as chemical vapor deposition, physical vapor deposition (e.g., evaporation or sputtering), or epitaxial techniques, among other deposition processes. Processes for fabricating circuit elements described herein can entail the removal of one or more materials from a device during fabrication. Depending on the material to be removed, the removal process can include, e.g., wet etching techniques, dry etching techniques, or lift-off processes. The materials forming the circuit elements described herein can be patterned using known lithographic techniques (e.g., photolithography or e-beam lithography). During operation of a quantum computational system that uses superconducting quantum circuit elements and / or superconducting classical circuit elements, such as the circuit elements described herein, the superconducting circuit elements are cooled down within a cryostat to temperatures that allow a superconductor material to exhibit superconducting properties. A superconductor (alternatively superconducting) material can be understood as material that exhibits superconducting properties at or below a superconducting critical temperature. Examples of superconducting material include aluminum (superconductive critical temperature of 1.2 kelvin) and niobium (superconducting critical temperature of 9.3 kelvin). Accordingly, superconducting structures, such as superconducting traces and superconducting ground planes, are formed from material that exhibits superconducting properties at or below a superconducting critical temperature. In certain implementations, control signals for the quantum circuit elements (e.g., qubits and qubit couplers) may be provided using classical circuit elements that are electrically and / or electromagnetically coupled to the quantum circuit elements. The control signals may be provided in digital and / or analog form. Attorney Docket No. 56113-0503WO1 Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto- optical disks; CD-ROM and DVD-ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantum memories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence. Control of the various systems described in this specification, or portions of them, can be implemented in a computer program product that includes instructions that are stored on one or more non-transitory machine-readable storage media, and that are executable on one or more processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or system that may include one or more processing devices and memory to store executable instructions to perform the operations described in this specification. While this specification contains many specific implementation details, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination. Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the described program components and Attorney Docket No. 56113-0503WO1 systems can generally be integrated together in a single software product or packaged into multiple software products. Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. What is claimed is:

Claims

Attorney Docket No. 56113-0503WO1 CLAIMS 1. A method for performing an electronic structure calculation, the method comprising: preparing, by quantum computation, approximations of a ground state in an active space of a quantum system; measuring, by quantum computation, the approximations of the ground state to obtain a plurality of cluster amplitudes; and processing, by classical computation, the plurality of cluster amplitudes to simulate the quantum system, comprising providing the plurality of cluster amplitudes as input to a coupled cluster method that recovers dynamic correlation in the quantum system.

2. The method of claim 1, wherein processing the plurality of cluster amplitudes to simulate the quantum system comprises solving coupled cluster equations to calculate an energy or set of observables for the quantum system.

3. The method of claim 2, wherein solving coupled cluster equations to calculate an energy or set of observables for the quantum system comprises: fixing values of the plurality of cluster amplitudes in the coupled cluster equations to the measured plurality of cluster amplitudes; solving for remaining cluster amplitudes in the coupled cluster equations to obtain a coupled cluster wavefunction; and calculating the energy or set of observables for the quantum system using the coupled cluster wavefunction.

4. The method of any one of claims 1 to 3, wherein measuring the approximations of the ground state to obtain a plurality of cluster amplitudes comprises performing a polynomial number of measurements to a predetermined fixed error.

5. The method of any one of claims 1 to 4, wherein the plurality of cluster amplitudes comprise computational basis amplitudes.

6. The method of claim 1, wherein the plurality of cluster amplitudes comprise configuration interaction coefficients in the active space.Attorney Docket No. 56113-0503WO1 7. The method of any one of claims 1 to 6, wherein preparing the approximations of the ground state in the active space of the quantum system comprises performing phase estimation or imaginary time evolution.

8. The method of any one of claims 1 to 7, wherein measuring the approximations of the ground state to obtain the plurality of cluster amplitudes comprises performing state tomography, shadow tomography, or sign tomography with sampling.

9. The method of any one of claims 1 to 8, wherein the coupled cluster method comprises a tailored coupled cluster method.

10. The method of claim 9, wherein the cluster amplitudes comprise cluster amplitudes up to second order in active space.

11. The method of any one of claims 1 to 8, wherein the coupled cluster method comprises an externally-corrected coupled cluster method.

12. The method of claim 11, wherein the cluster amplitudes comprise cluster amplitudes up to fourth order.

13. The method of any one of claims 1 to 11, wherein the quantum system comprises a quantum system with electronic structure, for example a chemical or material.

14. An apparatus comprising classical computing hardware in communication with quantum computing hardware, wherein the apparatus is configured to perform operations according to the method of any one of claims 1 to 13.

15. The apparatus of claim 14, wherein the quantum hardware comprises a noisy intermediate scale quantum computer.

16. The apparatus of claim 14, wherein the quantum hardware comprises a fault tolerant quantum computer.

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