Execution of unbiasing fermionic quantum monte carlo with quantum computer and shadow tomography

Abstract

To provide a method, a system and an apparatus for hybrid quantum classical quantum Monte Carlo.SOLUTION: In a system 100, a classical processor executes a method including the steps of: receiving data indicating a result of measurement of a trial wave function approximate to a target wave function which is generated by a quantum processor; calculating a classical shadow of the trial wave function using the received data; and executing imaginary time propagation for a sequence of an imaginary number time step of an initial wave function by using Hamiltonian characterizing a Fermionic quantum system. In the step of executing the imaginary number time step, the imaginary number time propagation is executed until a predetermined convergence reference is satisfied. The classical shadow of the trial wave function is used to update a wave function for the previous imaginary number time step, and thereby a wave function for the current imaginary number time step is acquired.SELECTED DRAWING: Figure 1

Description

[Technical Field]

[0001] This specification relates to quantum computing. [Background technology]

[0002] Computing an accurate solution to the Schrödinger equation for the ground state of many-electron systems has applications across nearly every field of modern science, enabling a detailed understanding of important unsolved questions in chemistry, physics, materials science, and biology. However, the complexity of the Schrödinger equation grows exponentially with the number of electrons in the system. Therefore, progress towards an efficient means to accurately compute the ground-state quantum mechanical properties of complex systems has been slow.

[0003] Known general-purpose methods for computing solutions to the Schrödinger equation can be grouped into two categories. The first category includes methods that scale exponentially with system size and produce numerically exact answers. The second category includes methods that incur the cost of scaling polynomially with system size and rely on canceling errors when computing observables. Techniques in the second category are currently the only methods that can be feasibly applied to large systems, but the accuracy of the solutions obtained in such cases is unsatisfactory and almost always difficult to utilize.

[0004] Quantum computing offers an alternative computational paradigm that can complement and potentially surpass classical methods in efficiency. In the absence of fault-tolerant quantum computers, noisy intermediate-scale quantum computing (NISQ) techniques can be used to explore many-body quantum problems. NISQ algorithms for quantum ground-state computation are largely centered around the variational quantum eigensolver (VQE) framework, which requires addressing optimization problems and noisy gradients. As an alternative, algorithms based on imaginary time evolution methods have been proposed that in principle avoid the optimization problem. However, due to the nonunitary nature of imaginary time evolution methods, optimization heuristics must be used to achieve reasonable scaling in system size. Therefore, alternative computational strategies that circumvent these limiting factors are needed to enable the first practical quantum supremacy in fermionic simulations. Summary of the Invention [Means for solving the problem]

[0005] This paper describes a hybrid quantum-classical algorithm for implementing unbiased fermionic quantum Monte Carlo using quantum computers and shadow tomography.

[0006] In general, one inventive aspect of the subject matter described herein can be implemented in a method for performing a quantum Monte Carlo simulation of a fermionic quantum system to calculate a target wave function of the fermionic quantum system, the method including: receiving, by a classical computer, data generated by the quantum computer, the data representing results of one or more measurements of a transformed trial wave function, the trial wave function approximating the target wave function and prepared by the quantum computer; calculating, by the classical computer, a classical shadow of the trial wave function using the data representing the results of the one or more measurements of the transformed trial wave function; and performing, by the classical computer, imaginary time propagation for a sequence of imaginary time steps of an initial wave function using a Hamiltonian characterizing the fermionic quantum system, the imaginary time propagation being performed until a predetermined convergence criterion is met, wherein performing each imaginary time step of the imaginary time propagation includes using the classical shadow of the trial wave function to update the wave function for a previous imaginary time step to obtain the wave function for a current imaginary time step.

[0007] Other implementations of these aspects include 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 consisting of one or more classical and / or 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, during operation, causes 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 a data processing device, cause the apparatus to perform the actions.

[0008] Each of the above and other implementations may optionally include one or more of the following features, alone or in combination: In some implementations, updating the wave function for the previous imaginary time step using the classical shadow of the trial wave function includes determining a Walker wave function for the current imaginary time step, and determining Walker weights for the current imaginary time step using a first dot product of the trial wave function and the Walker wave function for the previous imaginary time step and a second dot product of the trial wave function and the Walker wave function for the current imaginary time step, where the first dot product and the second dot product are determined using the classical shadow of the trial wave function.

[0009] In some implementations, the method further includes storing the calculated classical shadow of the trial wave function in a classical memory of a classical computer.

[0010] In some implementations, determining the Walker weights for the current imaginary time step using a first dot product of the trial wave function and the Walker wave function for the previous imaginary time step and a second dot product of the trial wave function and the Walker wave function for the current imaginary time step includes retrieving a classical shadow of the trial wave function from classical memory; computing an approximation of the first dot product, which includes determining expectation values ​​of one or more classically simulated first projectors and classical shadows of the trial wave function, where the one or more first projectors depend on the Walker wave function for the previous imaginary time step; and computing an approximation of the second dot product, which includes determining expectation values ​​of one or more classically simulated second projectors and classical shadows of the trial wave function, where the one or more second projectors depend on the Walker wave function for the current imaginary time step.

[0011] In some implementations, the one or more first projectors are generated using a stabilizer state.

[0012] In some implementations, the stabilizer state includes a computational basis state with a Hamming weight equal to the number of particles represented by the trial state.

[0013] In some implementations, the transformed trial wavefunction comprises a trial wavefunction rotated with a unitary operator randomly sampled from an ensemble of unitaries, where the ensemble of unitaries is tomographically complete. In some implementations, the unitary operator comprises an N-qubit Clifford circuit or a tensor product of randomly selected Clifford circuits for fewer than N qubits.

[0014] In some implementations, performing each imaginary time step of the imaginary time propagation further comprises calculating an energy estimate using a classical shadow of the trial wave function.

[0015] In some implementations, the transformed trial wavefunctions include trial wavefunctions transformed using a tensor product of unitary operators, where each unitary operator in the tensor product is a respective randomly selected N p∈P Including the Qubit Clifford gate, N p∈P represents the number of qubits in part p of the division of N qubits into P parts.

[0016] In some implementations, the quantum Monte Carlo simulation includes a projector quantum Monte Carlo simulation or an auxiliary field quantum Monte Carlo simulation.

[0017] In some implementations, the quantum computer includes a noisy intermediate-scale quantum device.

[0018] In some implementations, the trial wave function includes a wave function from the generalized valence bond perfect pairing wave function hypothesis.

[0019] In some implementations, the generalized valence bond perfect pairing wave function hypothesis includes a first set of layers containing density-density terms and a second set of layers containing nearest-neighbor hopping terms between the same spin pairs.

[0020] The subject matter described herein can be implemented in a particular manner to realize one or more of the following advantages.

[0021] Systems implementing the techniques described herein can target quantum states and their properties with increased computational efficiency and increased precision. For example, in our hybrid quantum-classical quantum Monte Carlo algorithm, classically implemented quantum Monte Carlo methods do not need to repeatedly query a quantum computer. This separation of interactions between quantum and classical computers avoids the need to minimize latency, which is a particularly attractive feature on the NISQ platform.

[0022] Furthermore, systems implementing the techniques described herein use trial wavefunctions that are inherently more accurate than traditional trial wavefunctions, e.g., single determinants, and that can be obtained by efficient polynomial-scaling classical methods that circumvent the problems of variational optimization in quantum computers. The trial wavefunctions may include wavefunctions for which no known polynomial-scaling classical algorithms exist for the evaluation of quantities required by quantum Monte Carlo methods. The trial wavefunctions achieve polynomial scaling, and thus the techniques described herein achieve exponential computational speedup compared to classical counterparts.

[0023] Furthermore, systems implementing the techniques described herein can compute quantities required by quantum Monte Carlo methods, such as wave function overlaps, with a bounded number of experimental and measurement iterations (without constraints on the shape of the trial wave functions).

number

[0024] Furthermore, the techniques described herein are robust to noise, e.g., noise resulting from hardware failure, since the directly calculated quantity is the ratio between overlap values, which is inherently robust to overlap being rescaled by some error channel.

[0025] The details of one or more implementations of the subject matter herein 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, drawings, and claims. [Brief explanation of the drawings]

[0026] [Figure 1] FIG. 1 is a block diagram of an exemplary system implementing a hybrid quantum-classical QMC algorithm. [Figure 2] 1 is a flow diagram of a first exemplary process for performing a quantum Monte Carlo simulation of a fermionic quantum system to calculate a target wave function and / or properties of the target wave function of the fermionic quantum system using shadow tomography. [Figure 3] FIG. 1 illustrates the application of the QC-QMC algorithm described herein to the H4 molecule in an eight-qubit experiment. [Figure 4] 10 is a flow diagram of a second exemplary process for performing a quantum Monte Carlo simulation of a fermionic quantum system to calculate a target wave function and / or properties of the target wave function of the fermionic quantum system. [Figure 5] FIG. 1 illustrates an exemplary classical / quantum computer. DETAILED DESCRIPTION OF THE INVENTION

[0027] Quantum Monte Carlo (QMC) methods are based on the many-body Hamiltonian

number

number

number

[0028] the ground state energy of the target state, i.e., E ground = E(τ=∞) can be estimated by averaging the time series of {〈E(τ)〉} given by the weighted average over M statistical samples,

number

[0029] Although formally rigorous, such stochastic imaginary time evolution algorithms typically encounter the well-known fermionic sign problem, which manifests itself in the alternating sign in the weights of each statistical sample. In the worst case, the fermionic sign problem causes energy estimators to have exponentially large variance, entailing the need to average exponentially many samples to obtain a fixed-precision estimate of an observable, such as the ground-state energy. Therefore, reliable computation of the ground state and its properties is practically infeasible, and rigorous unbiased QMC methods are only applicable to small systems or those free of the sign problem.

[0030] In the first quantized QMC method, this problem renders itself as a bosonic ground state. Because fermionic antisymmetry is not explicitly imposed, the true ground state of the first quantized Hamiltonian is in fact bosonic. This then requires imposing a fermionic node structure in the first quantization to calculate the fermionic ground state. In the second quantized QMC method, the bosonic state cannot be obtained from the fermionic Hamiltonian. The sign problem manifests itself differently. Statistical estimates from the second quantized QMC method exhibit variance that grows exponentially with system size.

[0031] The sign problem is that each wave function, i.e., |φ i By imposing constraints on the imaginary time evolution of each statistical sample, denoted by |(τ)〉, it can be controlled to give an estimate of the ground state energy with a polynomially bounded variance. These constraints, e.g., fixed node and phaseless approximations, can be applied to the trial wavefunction |Ψ T >, the accuracy of constrained QMC depends on the choice of trial wavefunction. Such constraints inevitably introduce a potentially significant bias into the final ground-state energy estimate.

[0032] Classically, computationally tractable choices for trial wave functions are limited to states such as a single mean-field determinant, e.g., a Hartree-Fock state, a linear combination of mean-field states, a simple form of an electron-electron pair (two-body) correlator (commonly called a Jastrow factor) applied to the mean-field state, or some other physically induced transformation applied to the mean-field state, such as the backflow technique. On the other hand, wave functions that can be prepared with quantum circuits are candidates for trial wave functions on quantum computers, including more general two-body correlators. These trial wave functions are referred to herein as "quantum" trial wave functions.

[0033] This paper describes a hybrid quantum-classical QMC algorithm that combines constrained quantum Monte Carlo (QMC) with quantum computing techniques to reduce bias in the final quantum state estimate. The hybrid quantum-classical QMC algorithm (QC-QMC) uses quantum trial wave functions while implementing most of the imaginary time evolution method on a classical computer. That is, the classical computer calculates the quantum trial wave functions for each statistical sample |φ i (τ)〉 and estimate the ground state energy E (i) (τ). During this procedure, constraints from the quantum trial wavefunction are imposed to control the sign problem.

[0034] To implement constrained time evolution, the only primitive function that requires a quantum computer is the trial wave function |Ψ T 〉 and the statistical sample wave function |φ at any imaginary time τ i(τ)〉. In particular, the QC-QMC algorithm described herein uses shadow tomography to estimate the overlap between the trial wave function and the statistical sample. Based on experiments, this involves performing a set of randomly chosen measurements of a reference state associated with the trial wave function prior to initiating the QMC method. This allows for efficient estimation of the entire set of required overlaps using a small number of experimental iterations combined with classical post-processing. It is not necessary for the classically implemented QMC method to repeatedly query the quantum computer in this QC-QMC formulation, even though the details of the statistical sample are not determined in advance. Separating the interaction between the quantum and classical computers avoids the need to minimize latency, a particularly attractive feature on the NISQ platform.

[0035] While the QC-QMC algorithm described herein generally applies to any form of constrained QMC, for illustrative purposes, this specification describes a specific demonstration of the QC-QMC algorithm using an implementation of QMC known as auxiliary field QMC (AFQMC). AFQMC is a PQMC method that operates in the second quantization space. Thus, the sign problem in AFQMC manifests itself in increasing the variance in statistical estimates. To impose constraints on the imaginary time propagation, a trial wave function is introduced that can be used in importance sampling as well as constraints. As a result, the wave function in imaginary time τ is

number

[0036] In some implementations, the Walker wave function in Eq. 3 can be chosen to be a single Slater determinant, and the imaginary propagation for small time steps Δτ in Eq. 1, i.e.

number

[0037] While iteratively applying imaginary time propagation to the wavefunction, the AFQMC algorithm ensures that all weights remain true and positive, resulting in a final energy estimator, i.e.

number

number

number

[0038] Currently, classically tractable trial wave functions are either single determinants or linear combinations of determinants. The former are scalable (up to around 500 electrons) but can often be inaccurate, especially for strongly correlated systems. The latter are limited to small numbers of electrons (around 14) but can be very accurate even for strongly correlated systems. The choice of trial wave function in AFQMC is constrained by the evaluation of Equation 4 and Equation 6. Computation of either one of these scales exponentially with system size, and the resulting AFQMC method is exponentially expensive.

[0039] The QC-QMC algorithm described herein uses a class of trial wavefunctions that are inherently more accurate than a single determinant and can be obtained by efficient polynomial-scaling classical methods, circumventing the problems of variational optimization on quantum computers. The trial wavefunctions may include wavefunctions for which no known polynomial-scaling classical algorithm exists for the evaluation of Equation 4 and Equation 6. Quantum computers are used to remove such limitations by introducing polynomial-scaling algorithms for Equation 4 and Equation 6, which guarantees exponential speedup compared to classical computers. In the QC-QMC algorithm described herein, Equation 4 and Equation 6 can be measured on a quantum computer, and the actual imaginary time propagation can be implemented classically. This separates the subroutines into those that need to be run on a quantum computer and those that need to be run on a classical computer.

[0040] In some implementations, the trial wave function can be a variant of the coupled cluster wave function, which has an exponential parameterization, i.e.

number

number

number

number

[0041] The use of CCSD (or other higher-order coupled cluster wave functions) is not suitable for use as AFQMC trial wave functions, since the projection of such functions onto arbitrary Slater determinants cannot be efficiently computed without approximations. This is true for nearly all non-trivial variants of coupled clusters. Generalized valence bond perfect pairing (PP) 1,2 The cost of computing wavefunction overlaps for coupled-cluster methods with a limited set of amplitudes, such as , also scales exponentially with system size. The required overlaps of such wavefunctions can be efficiently evaluated by using a quantum computer to prepare unitary versions of, or approximations to, coupled-cluster wavefunctions. The use of classically optimizable coupled-cluster wavefunctions avoids costly variational optimization procedures in quantum devices.

[0042] An exemplary coupled-cluster wave function hypothesis that can be used as a trial wave function is the generalized valence bond PP hypothesis.

number

number

number

[0043] PP wave functions often fall short in achieving qualitative accuracy. This is best demonstrated in systems where pairwise correlations, such as multiple bond breaking, become important. While there are several ways to classically incorporate these pairwise correlations, we describe here that multiple layers of QC-QMC hardware-efficient operators can be added to the PP ansatz. There are two types of these additional layers that can be added: 1. The first type of layer is the density-density product term

number

[0044] Note that all operators in this layer are commutative with each other so that there is no Trotter error. 2. The second kind is the "nearest neighbor" hopping term between pairs of the same spin (σ)

number

[0045] Multiple layers of each type can be alternately applied to the PP hypotheses to improve the overall accuracy. The effectiveness of these layers varies with the choice of i, j pairs.

[0046] 1 is a block diagram of an exemplary system 100 that implements the QC-QMC algorithms described herein. System 100 is an example of a system that can implement the systems, components, and techniques described below, implemented as quantum and classical computer programs on quantum computing devices and classical computers at one or more locations.

[0047] Exemplary system 100 includes a quantum processor 102 in data communication with a classical processor 104. For illustrative purposes, quantum processor 102 and classical processor 104 are shown as separate entities, although in some implementations classical processor 104 may be included within quantum processor 102.

[0048] Quantum processor 102 includes components for performing quantum computations. For example, quantum processor 102 may include a qubit array, quantum circuitry, and a control device configured to manipulate physical qubits in the qubit array and apply the quantum circuitry to the qubits. An exemplary quantum processor is described in more detail below with reference to FIG. 5.

[0049] Classical processor 104 includes components for performing classical computations. For example, classical processor 104 may be configured to send data specifying a trial wave function to quantum processor 102 and to receive data representing results of measurement operations performed by quantum processor 102. Classical processor 104 may be further configured to process the received data representing results of measurement operations performed by quantum processor 102 to compute a classical representation of a target state or properties of the target state.

[0050] As mentioned above, the QC-QMC algorithm described in this application implements the QMC imaginary time evolution method using shadow tomography. Shadow tomography is a process that can be used to estimate the properties of quantum states without resorting to full-state tomography. Let ρ denote some unknown quantum state. It is assumed that we have access to N copies of ρ. {O i Let} denote a collection of M observables. A task is i For a certain amount of additive error ε, the quantity Tr(ρO i ) This can be done efficiently in some situations by randomly choosing measurement operators from a tomographically complete set, i.e., a set that forms an operator basis on the Hilbert space of the system.

[0051] To specify a protocol, an ensemble of unitaries u is chosen. Then, the unitary U k ∈u is randomly sampled and the state

number

number

number

[0052] M is required to be invertible, which holds if and only if the collection of measurement operators defined by extracting U∈u and measuring in the computational basis is tomographically complete. Assuming this holds, M -1 can be applied to both sides of Equation 14,

number

number

[0053] Thus, in step (A) of the QC-QMC algorithm, quantum processor 102 performs a randomly chosen set of measurements on copies of the trial wave function for the QMC method.

number

[0054] For each experiment in the plurality of experiments, quantum processor 102 may apply a quantum circuit to the physical qubits contained within quantum processor 102. The circuit may include a first circuit that prepares the qubits in an initial state, e.g., a superposition of a trial wave function and a zero state, and a second quantum circuit that implements a measurement operator for a shadow tomography experiment. The specific forms of the first and second circuits depend on the trial wave function used.

[0055] As an example, in an implementation where the trial wave function is a perfect pairing state (PP), the first circuit generates the quantum state |τ〉=(|0〉+|Ψ TIn this example, it is sufficient to prepare the quantum state (|0〉 + |PP(θ)〉) / √2, where |PP(θ)〉 represents the complete pairing state with a vector of state parameters θ,

number

number

[0056] In this example, for the second quantum circuit, the measurement operator is

number

number

[0057] In some implementations, the following global stabilizer measurement strategy can be implemented to reduce the size of the quantum circuitry required to perform shadow tomography. In general, we apply a unitary U and then compute the basis {|x〉:x∈{0,1} n}, the measurement as originally proposed by shadow tomography is based on the rotational basis {U † |x〉:x∈{0,1} n For a set of unitaries u, choosing a unitary from it uniformly at random and then measuring it on a computational basis is equivalent to measuring it on a POVM.

number

number

number

number

number

number

number

[0058] F n Let be the “H-free” group for n qubits, i.e., the group generated by X, CNOT, and CZ. The action of any H-free operator is

number

number

number

number

number

number

[0059] Furthermore, in some implementations, a piecewise shadow tomography strategy may be implemented to reduce quantum circuit depth, which is described in more detail below with reference to FIG.

[0060] In step (C) of the QC-QMC algorithm, the classical processor processes the received measurements and calculates the classical shadow, which can be stored in the classical memory 106 of the classical processor 104.

[0061] In stage (D) of the QC-QMC algorithm, classical processor 104 uses the stored classical shadows to implement the QMC algorithm. That is, classical processor 104 performs imaginary time propagation for a sequence of imaginary time steps of an initial wave function using a Hamiltonian that characterizes the fermionic quantum system, e.g., according to Equation 1. At each imaginary time step, classical processor 104 uses the stored classical shadows to calculate the required wave function overlap. Exemplary operations performed by classical processor 104 are described in more detail below with reference to FIG. 2.

[0062] In stage (E) of the QC-QMC algorithm, classical processor 104 outputs data representing a target quantum state. In some implementations, classical processor 104 can use the data representing the target quantum state to calculate properties of the target quantum state, for example, the expected energy of the target quantum state, as described above with respect to Equations 2 and 4-6.

[0063] 2 is a flow diagram of an exemplary process 200 for performing a quantum Monte Carlo simulation of a fermionic quantum system to calculate a target wave function and / or properties of the target wave function, e.g., the ground state energy, of the fermionic quantum system. For convenience, process 200 is described as being performed by a system including classical and quantum computing devices at one or more locations. For example, system 100 of FIG. 1 , suitably programmed in accordance with this specification, can perform process 200.

[0064] The system uses a quantum computing device to prepare multiple copies of a trial wave function (step 202). The trial wave function is a wave function that approximates the target wave function. In some implementations, the trial wave function can be, for example, a wave function from the generalized valence bond perfect pairing wave function hypothesis, which includes a first set of layers containing density-density product terms and a second set of layers containing nearest-neighbor hopping terms between the same spin pairs.

[0065] The system uses a quantum computing device to perform a measurement operation on multiple copies of the trial wave function (step 204). In some implementations, to perform the measurement operation, the quantum computing device generates a transformed trial wave function by rotating the trial wave function with a unitary operator randomly sampled from an ensemble of unitaries, where the ensemble of unitaries is tomographically complete. The unitary operator used may be a tensor product of an N-qubit Clifford circuit or randomly selected Clifford circuits for less than N qubits. For example, the transformed trial wave function may be, as described above in the discussion surrounding Equation 14,

number

[0066] In some implementations, the system can partition qubits included in the quantum computing device as described below with respect to Equations 29-34. In these implementations, the system can transform the trial wave function by applying a tensor product of unitary operators to the trial wave function, where each unitary operator in the tensor product is a respective one of N randomly selected unitary operators. p∈P Cubit Clifford gate, N p∈P represents the number of qubits in portion p of the partition of N qubits into P portions, as described below with respect to Equations 29-34. The quantum computing device can then measure the transformed trial wavefunctions with a computational basis to obtain respective measurement results.

[0067] The system transmits data representing the results of the measurement operation from the quantum computing device to a classical computing device included in the system (step 206).

[0068] The classical computing device receives data representing the results of measuring the transformed trial wave function produced by the quantum computing device and uses this data to calculate a classical shadow of the trial wave function (step 208). The calculation of the classical shadow is described above with respect to Equations 13 and 14. The classical computing device can efficiently store the calculated classical shadow in classical memory of the classical computing device.

[0069] The system uses a classical computing device to perform imaginary time propagation (for a sequence of imaginary time steps) of an initial wave function using a Hamiltonian that characterizes the fermionic quantum system (step 210). The imaginary time propagation may be performed until a predetermined convergence criterion is met, for example, until the output wave function converges to within a predetermined threshold, which may depend on the target accuracy.

[0070] At each imaginary time step of the imaginary time propagation, the classical computing device updates the wave function for the previous imaginary time step using the classical shadow of the trial wave function to obtain the wave function for the current imaginary time step. To update the wave function for the previous imaginary time step using the classical shadow of the trial wave function, the classical computer determines a Walker wave function for the current time step, e.g., through imaginary time propagation, and determines Walker weights for the current time step using i) a first dot product of the trial wave function and the Walker wave function for the previous time step and ii) a second dot product of the trial wave function and the Walker wave function for the current time step, where the first dot product and the second dot product are determined using the classical shadow of the trial wave function.

[0071] Exemplary techniques implemented by the system to determine the dot product of the trial wave function and the Walker wave function using classical shadows include: |Ψ TLet |Ψ denote the trial wave function. In some implementations, |Ψ T 〉 can be chosen to represent a fermionic wave function with a certain number of particles η>0, and we can use the quantum state encoded in the Jordan-Wigner transformation, so that |Ψ T The qubit wave function for 〉 is a superposition of the computational basis states and the Hamming weight η.

[0072] Let |φ〉 denote the Walker wave function, which may be a superposition of the computational basis state and the Hamming weight η. Computing the dot product of a trial wave function and a Walker wave function is therefore done by using the classical shadow of the trial wave function to calculate the dot product 〈φ|Ψ T 〉.

[0073] When the quantum computing device prepares a copy of the trial wave function in step 202 of exemplary process 200, the quantum computing device may prepare a quantum state |τ〉〈τ|, where:

number

number

number

[0074] If the ensemble u used to generate the classical shadow is a Clifford group for N qubits, then the inverse of the channel M can be written as

number

number

number

number

[0075] The overlap between stabilizer states (including the basis states) can be calculated classically and efficiently using the Gottsman-Knill theorem, and the right-hand side of Eq. 28 can be calculated classically and efficiently. In particular, 〈b k |U k |0〉 is the Clifford circuit U k The Walker wave function can be written as a linear combination of polynomial stabilizer states, so the quantity

number

[0076] As mentioned above, shadow tomography using N-qubit Clifford groups can be used to simultaneously estimate M quantities, such as those in Equation 27, with a cost that scales logarithmically with M. However, implementing these measurements on NISQ devices can be challenging due to the required circuit depth. An alternative preference for an ensemble of random unitaries u can alleviate this issue. A second preference for u involves a unitary U∈u chosen to be a tensor product of single-qubit Clifford operators. It is also possible to interpolate between these two extremes. It can be shown that the choice of single-qubit Clifford for U leads to a limit on the cost of shadow tomography that scales exponentially with the locality of the estimated operator. Projectors are highly nonlocal operators, and therefore one can expect to encounter a large number of measurement iterations required when using single-qubit Clifford shadow tomography to estimate their expectation values ​​(assuming practical implementation correlates with the limit). This suggests that a compromise between circuit depth and the number of iterations required to perform shadow tomography with different preferences for U should be considered.

[0077] To that end, we use u consisting of a randomly sampled tensor product of Clifford unitaries for fewer than N qubits, where 〈β|ΨT To efficiently perform the classical post-processing required to estimate 〉, an alternative technique can be implemented. The expression in Equation 28 can also be written as:

number

number

number

number

number

[0078] In some implementations, the partition may include two parts, one for each spin sector. In implementations where the Walker wave function is a superposition of a basis state with Hamming weight η and a non-zero number of electrons in each spin sector, the shadow tomography is a function of the Walker wave function and the p It may be used to evaluate the overlap of trial wave functions where |0〉 = 0. Thus, the dot product is

number

[0079] Returning to step 210 of FIG. 2 , to determine the Walker weights for the current time step using the first dot product of the trial wave function and the Walker wave function for the previous time step and the second dot product of the trial wave function and the Walker wave function for the current time step, the classical computer retrieves from classical memory the classical shadow of the trial wave function, e.g., retrieves data corresponding to Equation 14. The classical computer then calculates an approximation of the first dot product by determining an expectation value between one or more classically simulated first projectors and the classical shadow of the trial wave function, e.g., the expectation value given by Equation 28. The one or more first projectors depend on the Walker wave function for the previous time step. That is, the classical computer calculates an approximation of the first dot product〈Ψ〉 using Equations 19-28 or 29-34. T |Ψ n (τ)〉. As described above with respect to Equations 19-28, one or more first projectors can be generated using stabilizer states, where the stabilizer states include computational basis states with Hamming weights equal to the number of particles represented by the trial wave function. A similar operation can be performed by a classical computer to compute the second dot product 〈Ψ T |Ψ n This can be implemented to calculate (τ+Δτ)〉.

[0080] At each imaginary time step of the imaginary time propagation, the classical computer also computes an energy estimate, e.g., given by Equation 3, using a classical shadow of the trial wave function. In some implementations, the ground state energy is estimated from the time series of energy estimates computed at each imaginary time step.

[0081] Figure 3 illustrates the application of the QC-QMC algorithm described herein to the H4 molecule in an eight-qubit experiment. In this example, an eight-spin-orbit quantum trial wavefunction is used. The trial wavefunction consists of a valence bond wavefunction known as the perfect pairing state, to which a hardware-efficient quantum circuit with offline single-particle rotations is applied. This would classically be inconvenient as a trial wavefunction for AFQMC.

[0082] Part (a) of Figure 3 shows an exemplary state preparation circuit for preparing a trial wave function using a quantum computer. This 8-qubit experiment considers H4 in a square with side length 1.23 A and dissociating into four hydrogen atoms. This system can be used as a testbed for electron correlation methods in quantum chemistry. Part (a) shows the experimental circuit used for experiments across a 2 × 4 qubit grid. In the circuit diagram, H denotes a Hadamard gate, G denotes a Givens rotation gate (generated by the Pauli gate (XX + YY)), P denotes a Pauli gate, and |Ψ T 〉 denotes the quantum trial wavefunction. Offline orbital rotations are not present in the actual quantum circuit, as they can be handled efficiently by classical post-processing.

[0083] Parts (b) and (c) of Figure 3 show the convergence of the atomization energy of H4 as a function of the number of measurements. Part (b) shows a minimal basis set (STO-3G) with a total of four orbitals from four separate experiments with different sets of random measurements, and part (c) shows a quadruple zeta basis set (cc-pVQZ) with a total of 120 orbitals from two separate experiments. Different symbols in (b) and (c) indicate separate experimental results. The upper panels of (b) and (c) expand the energy range near the exact answer. As shown, noise in quantum devices can cause the quality of quantum trials to deviate from that of an ideal (i.e., noise-free) hypothesis, resulting in errors of as much as 10 kcal / mol in the atomization energy. Nevertheless, the QC-AFQMC described herein significantly reduces this error and achieves chemical precision in both bases. To further elucidate the QC-AFQMC results for H4, parts (b) and (c) show the trial evolution and QC-AFQMC energy as a function of the number of measurements performed on the device. Despite the presence of significant noise in the nearly 10 measurements, QC-AFQMC achieves chemical precision while addressing substantial residual bias in the underlying quantum trials.

[0084] 4 is a flow diagram of an exemplary process 400 for performing a quantum Monte Carlo simulation of a fermionic quantum system to calculate a target wave function and / or properties of the target wave function, e.g., basis wave function energies, of the fermionic quantum system. In some implementations, the quantum Monte Carlo simulation may be a projector quantum Monte Carlo simulation, e.g., an auxiliary field quantum Monte Carlo simulation. For convenience, process 400 is described as being performed by a system including classical and quantum computing devices at one or more locations. For example, system 100 of FIG. 1 , suitably programmed in accordance with this specification, may perform process 400.

[0085] A classical computer included in the system performs imaginary time propagation (for a sequence of imaginary time steps) of the initial wave function using a Hamiltonian that characterizes the fermionic quantum system (step 402). The imaginary time propagation is performed until a predetermined convergence criterion is met, e.g., until the output converges to within a predetermined threshold.

[0086] Each imaginary time step of imaginary time propagation includes the following steps: The classical computer sends data representing the wave function for the previous imaginary time step to a quantum computer, e.g., a NISQ device (step 404). The quantum computer calculates a dot product using data representing the wave function for the previous wave function and a trial wave function that approximates the target wave function (step 406). Exemplary trial wave functions are described above with reference to FIG. 1.

[0087] The classical computer receives data representing the calculated dot product generated by the quantum computer (step 408) and uses the data representing the calculated dot product to update the wave function for the previous imaginary time step to obtain the wave function for the current imaginary time step (step 410). The classical computer may also calculate an energy estimate using a classical shadow of the trial wave function, e.g., Equation 3.

[0088] In some implementations, the classical computer updates the wave function for the previous imaginary time step by using data representing the calculated dot product to determine a Walker wave function for the current time step and using the calculated dot product to determine Walker weights for the current time step to obtain a wave function for the current imaginary time step, where the calculated dot product includes a first dot product of the trial wave function and the Walker wave function for the current time step and a second dot product of the trial wave function and the Walker wave function for the previous time step. That is, the classical computer updates the wave function for the previous imaginary time step using Equations 3-6, where the dot product is calculated by the quantum computer. In these implementations, the data representing the wave function for the previous imaginary time step sent from the classical computer to the quantum computer includes data representing the Walker wave function for the previous imaginary time step and data representing the calculated Walker wave function for the current imaginary time step (e.g., calculated by the classical computer via imaginary time propagation).

[0089] The quantum computer can then calculate an inner product using data representing the Walker wave function for the previous imaginary time step, data representing the calculated Walker wave function for the current imaginary time step, and the trial wave function.

[0090] The quantum computer can compute an inner product using a projected measurement on the trial wave function, where the projector of the projected measurement is generated using a stabilizer state. The stabilizer state can include a computational basis state with a Hamming weight equal to the number of particles represented by the trial wave function. The projector of the projected measurement can be determined by data representing the Walker wave function for the previous imaginary time step or data representing the computed Walker wave function for the current imaginary time step. The computation of the inner product and the projected measurement that can be performed by a quantum computer are described above with references.

[0091] 5 illustrates an exemplary classical / quantum computer 500 for implementing some or all of the classical and quantum operations described herein. The exemplary classical / quantum computer 500 includes an exemplary quantum computing device 502. The quantum computing device 502 is intended to represent various forms of quantum computing devices. The components shown, their connections and relationships, and their functions are merely exemplary and do not limit the implementation of the invention described and / or claimed herein.

[0092] Exemplary quantum computing device 502 includes a qubit assembly 552 and a control and measurement system 504. The qubit assembly includes a plurality of qubits, e.g., qubit 506, that are used to perform algorithmic operations or quantum computations. While the qubits shown in FIG. 5 are arranged in a rectangular array, this is a schematic representation and is not intended to be limiting. Qubit assembly 552 also includes adjustable coupling elements, e.g., couplers 508, that enable interaction between the coupled qubits. In the schematic representation of FIG. 5, each qubit is adjustably coupled to each of its four neighboring qubits using a respective coupling element. However, this is an example arrangement of qubits and couplers, and other arrangements are possible, including non-rectangular arrangements, arrangements that allow coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits.

[0093] Each qubit may be a physical two-level quantum system or device with levels representing logical values ​​of 0 and 1. The specific physical implementation of the qubits and how they interact with each other will depend on various factors, including the type of quantum computing device 502 included in exemplary computer 500 or the type of quantum computation the quantum computing device is performing. For example, in an atomic quantum computer, the qubits may be implemented by atomic, molecular, or solid-state quantum systems, e.g., hyperfine atomic states. As another example, in a superconducting quantum computer, the qubits may be implemented by superconducting qubits or semiconductor qubits, e.g., superconducting transmon states. As another example, in an NMR quantum computer, the qubits may be implemented by nuclear spin states.

[0094] In some implementations, quantum computation may proceed by, for example, loading qubits from a quantum memory and applying a sequence of unitary operators to the qubits. Applying the unitary operators to the qubits may include applying a corresponding sequence of quantum logic gates to the qubits to implement a quantum circuit required for shadow tomography, for example, as described above with reference to FIG. 1. Exemplary quantum logic gates include single-qubit gates, e.g., Pauli X, Pauli Y, Pauli Z (also referred to as X, Y, Z), Hadamard gates, S gates, rotations, two-qubit gates, e.g., controlled X, controlled Y, controlled Z (also referred to as CX, CY, CZ), controlled NOT gates (also referred to as CNOT), controlled swap gates (also referred to as CSWAP), iSWAP gates, and gates involving three or more qubits, e.g., Toffoli gates. Quantum logic gates can be implemented by applying control signals 510 generated by the control and measurement system 504 to the qubits and to the couplers.

[0095] For example, in some implementations, the qubits in qubit assembly 552 may be frequency tunable. In these examples, each qubit may have an associated operating frequency that can be adjusted by applying voltage pulses via one or more drive lines coupled to the qubit. Exemplary operating frequencies include a qubit idle frequency, a qubit interaction frequency, and a qubit readout frequency. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to a corresponding idle frequency may place the qubit in a state where it does not interact strongly with other qubits and can be used to perform a single-qubit gate. As another example, in cases where qubits interact through couplers with fixed couplings, the qubits may be configured to interact with each other by setting their respective operating frequencies at some gate-dependent frequency detuning from a common interaction frequency. In other cases, for example, when qubits interact through tunable couplers, the qubits can be configured to interact with one another by setting the parameters of each coupler to allow interaction between the qubits, and then setting the operating frequency of each of the qubits to a somewhat gate-dependent frequency that is detuned from their common interaction frequency. Such interactions may be implemented to implement multi-qubit gates.

[0096] The type of control signal 510 used depends on the physical implementation of the qubit. For example, the control signal may comprise an RF or microwave pulse in an NMR or superconducting quantum computer system, or an optical pulse in an atomic quantum computer system.

[0097] A quantum computation can be completed using a quantum observable, such as X or Z, by measuring the state of the qubit using each control signal 510. The measurement causes a readout signal 512 representing the measurement result to be communicated back to the measurement and control system 504. The readout signal 512 may include an RF, microwave, or optical signal, depending on the physical scheme for the quantum computing device and / or qubit. For convenience, the control signals 510 and readout signals 512 shown in FIG. 5 are shown as addressing only selected elements of the qubit assembly (i.e., the top and bottom rows), but in operation, the control signals 510 and readout signals 512 can address each element in the qubit assembly 552.

[0098] Control and measurement system 504 is an example of a classical computer system that can be used to perform various operations on qubit assembly 552, as described above, as well as other classical subroutines or calculations. Control and measurement system 504 includes one or more classical processors, e.g., classical processor 514, one or more memories, e.g., memory 516, and one or more I / O units, e.g., I / O unit 518, connected by one or more data buses. Control and measurement system 504 may be programmed to send sequences of control signals 510 to the qubit assembly, e.g., to perform a selected series of quantum gate operations, and to receive sequences of readout signals 512 from the qubit assembly, e.g., as part of performing a measurement operation.

[0099] The processor 514 is configured to process instructions for execution within the control and measurement system 504. In some implementations, the processor 514 is a single-threaded processor. In other implementations, the processor 514 is a multi-threaded processor. The processor 514 is capable of processing instructions stored in the memory 516.

[0100] The memory 516 stores information within the control and measurement system 504. In some implementations, the memory 516 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, the memory 516 may include a storage device capable of providing mass storage to the system 504, such as a hard disk device, an optical disk device, a storage device shared by multiple computing devices over a network (e.g., a cloud storage device), and / or some other mass storage device.

[0101] The input / output devices 518 provide input / output operations for the control and measurement system 504. The input / output devices 518 may include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers for sending control signals 510 to and receiving readout signals 512 from the qubit assemblies, as required by the physics scheme for the quantum computer. In some implementations, the input / output devices 518 may also include one or more network interface devices, e.g., Ethernet cards, serial communication devices, e.g., RS-232 ports, and / or wireless interface devices, e.g., 802.11 cards. In some implementations, the input / output devices 518 may include driver devices, e.g., keyboards, printers, and display devices, configured to receive input data and send output data to other external devices.

[0102] Although FIG. 5 illustrates an exemplary control and measurement system 504, implementations of the subject matter and functional operations described herein can be implemented in other types of digital electronic circuitry, or in computer software, firmware, or hardware including the structures disclosed herein and their structural equivalents, or in combinations of one or more of these.

[0103] The exemplary system 500 also includes an exemplary classical processor 550. The classical processor 550 can be used to perform the classical computing operations described herein according to some implementations.

[0104] Implementations of the subject matter and operations described herein may be implemented in digital electronic circuitry, analog electronic circuitry, suitable quantum circuitry, or more generally, in a quantum computing system, in tangibly embodied software or firmware, in computer hardware including the structures disclosed herein and their structural equivalents, or in a combination of one or more of these. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0105] Implementations of the subject matter described herein can be implemented as one or more computer programs, i.e., as one or more modules of computer program instructions encoded on a tangible, non-transitory storage medium for execution by or controlling the operation of a 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 these. Alternatively or additionally, the program instructions can be encoded on an artificially generated propagated signal capable of encoding digital and / or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal generated to encode digital and / or quantum information for transmission to a receiver device suitable for execution by a data processing apparatus.

[0106] The terms quantum information and quantum data refer to information or data carried by, held, or stored in quantum systems, with the smallest non-trivial system being a qubit, i.e., a system defining a unit of quantum information. It will be understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as two-level systems in the corresponding context. Such quantum systems may include, for example, multilevel systems with more than two levels. By way of example, such systems may include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the computational basis states are identified in ground and first excited states, although it will be understood that other setups are possible in which the computational states are identified in higher-level excited states.

[0107] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all types 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. An apparatus may 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 designed to simulate or generate 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. In addition to hardware, an apparatus may optionally include code that creates an execution environment for digital and / or quantum computer programs, e.g., code constituting processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of these.

[0108] Digital computer programs, which may be called or known as programs, software, software applications, modules, software modules, scripts, or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and may be deployed in any form, including as stand-alone programs or as modules, components, subroutines, or other units suitable for use in a digital computing environment. Quantum computer programs, which may be called or known as programs, software, software applications, modules, software modules, scripts, or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, or may be translated into a suitable quantum programming language, or may be written in a quantum programming language, e.g., QCL or Quipper.

[0109] A computer program may, but need not, correspond to a file in a file system. A program can be stored in file portions holding 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 cooperating files, e.g., files storing one or more modules, subprograms, or portions of code. A computer program may be deployed to run on one computer or multiple computers located at one location, or may be distributed across multiple locations and interconnected by a digital and / or quantum data communication network. A quantum data communication network is understood to be a network that can transmit quantum data using quantum systems, e.g., qubits. Generally, digital data communication networks cannot transmit quantum data, but quantum data communication networks can transmit both quantum data and digital data.

[0110] The processes and logic flows described herein may, where appropriate, be implemented by one or more programmable computers executing one or more computer programs in conjunction with one or more processors to perform functions by manipulating input data and generating output. The processes and logic flows may also be implemented by, and apparatus may be implemented as, special purpose logic circuitry, e.g., an FPGA or ASIC, or a quantum simulator, or a combination of special purpose logic circuitry or a quantum simulator with one or more programmed digital and / or quantum computers.

[0111] A system of one or more computers being "configured" to perform a particular operation or action means that the system has installed software, firmware, hardware, or a combination thereof that, when operated, causes the system to perform the operation or action. One or more computer programs being configured to perform a particular operation or action means that the one or more programs contain instructions that, when executed by a data processing device, cause the device to perform the operation or action. For example, a quantum computer may receive instructions from a digital computer that, when executed by a quantum computing device, cause the device to perform the operation or action.

[0112] A computer suitable for executing a computer program can be based on a general-purpose microprocessor or a special-purpose processor, or other type of central processing unit. Typically, the central processing unit receives instructions or data from a read-only memory, a random-access memory, or a quantum system suitable for transmitting quantum data, e.g., photons, or a combination thereof.

[0113] 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 memory may be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a computer also includes one or more mass storage devices for storing data, e.g., magnetic, magneto-optical, optical disks, or quantum systems suitable for storing quantum information, or is operatively coupled to the mass storage devices to receive data from, or transcribe data from, or both. A computer need not have such devices, however.

[0114] Quantum circuit elements (also called quantum computing circuit elements) include circuit elements for performing quantum processing operations. That is, quantum circuit elements are configured to utilize quantum mechanical phenomena such as superposition and entanglement to perform operations on data in a non-deterministic manner. Some quantum circuit elements, such as qubits, can be configured to represent and operate on information in multiple states 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-SQUIDs or DC-SQUIDs), among others.

[0115] In contrast, classical circuit elements generally process data in a deterministic manner. Classical circuit elements can be configured to collectively implement the instructions of a computer program by performing basic arithmetic, logic, and / or input / output operations on data, where the data is represented in analog or digital form. In some implementations, classical circuit elements can be used to send data to and / or receive data from 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 energy-efficient versions of RSFQ that do not use bias resistors.

[0116] In some cases, some or all of the quantum and / or classical circuit elements may be implemented using, for example, superconducting quantum and / or classical circuit elements. The assembly of superconducting circuit elements may involve the deposition of one or more materials, such as superconductors, dielectrics, and / or metals. Depending on the materials selected, these materials may 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. The processes for fabricating the circuit elements described herein may involve the removal of one or more materials from the device during assembly. Depending on the material to be removed, the removal process may include, for example, 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 electron beam lithography).

[0117] During operation of a quantum computing system using superconducting quantum and / or classical circuit elements, such as those described herein, the superconducting circuit elements are cooled in a cryostat to a temperature at which the superconducting material can exhibit superconducting properties. A superconductor (alternatively, superconducting) material can be understood as a material that exhibits superconducting properties below its superconducting critical temperature. Examples of superconducting materials include aluminum (superconducting critical temperature of 1.2 Kelvin) and niobium (superconducting critical temperature of 9.3 Kelvin). Thus, superconducting structures, such as superconducting traces and superconducting basal planes, are formed from materials that exhibit superconducting properties below their superconducting critical temperature.

[0118] In some implementations, control signals for 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.

[0119] Computer-readable media suitable for storing computer program instructions and data include, by way of example, all forms of non-volatile digital and / or quantum memories, media and memory devices, including semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices, magnetic disks, e.g., internal hard disks or removable disks, magneto-optical disks and CD-ROM and DVD-ROM disks, and quantum systems, e.g., trapped atoms or electrons. A quantum memory will be understood to be a device capable of storing quantum data for long periods of time with high fidelity and efficiency, e.g., a light-matter interface where light is used to transmit and matter is used to store and preserve the quantum characteristics of the quantum data, such as superposition or quantum coherence.

[0120] Control of the various systems described herein, or portions thereof, may be implemented in a computer program product that includes instructions stored on one or more non-transitory machine-readable storage media, the instructions being executable on one or more processing devices. The systems described herein, or portions thereof, may each be implemented as an apparatus, method, or system that may include one or more processing devices and memory for storing executable instructions for performing the operations described herein.

[0121] 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. Also, some features described herein in the context of separate implementations can be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable subcombination. Furthermore, while features may be described above as working in several combinations and initially claimed as such, one or more features from a claimed combination can, in some cases, be deleted from the combination, and the claimed combination may be directed to a subcombination or variations of the subcombination.

[0122] Similarly, while operations are shown in the figures in a particular order, this should not be understood as requiring that such operations be performed in the particular order or sequential order shown, or that all of the operations shown be performed, to achieve desirable results. In some situations, 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 systems may generally be integrated together in a single software product or packaged in multiple software products.

[0123] Specific implementations of the present 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 an example, the processes depicted in the accompanying figures do not necessarily require the particular order shown, or sequence, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous. [Explanation of symbols]

[0124] 100 systems 102 Quantum Processor 104 Classical Processors 106 Classical Memory 500 Classical / Quantum Computers, Computers, Systems 502 Quantum Computing Devices 504 Control and Measurement Systems, Systems 508 Combiner 514 Classical Processors, Processors 516 memory 518 I / O unit, input / output device 550 Classical Processor 552 Cubit Assembly

Claims

1. 1. A computer-implemented method for performing a quantum Monte Carlo simulation of a fermionic quantum system to calculate a target wave function of the fermionic quantum system, comprising: receiving, by a classical computer, data generated by a quantum computer, the data representing the results of one or more measurements of a trial wave function, the trial wave function approximating the target wave function, and prepared by the quantum computer; calculating, by the classical computer, a classical shadow of the trial wave function using the data representing the results of the one or more measurements of the trial wave function; performing, by the classical computer, imaginary time propagation for a sequence of imaginary time steps of an initial wave function using a Hamiltonian that characterizes the fermionic quantum system; The imaginary time propagation is performed until a predetermined convergence criterion is met; performing each imaginary time step of the imaginary time propagation includes using the classical shadow of the trial wave function to update the wave function for a previous imaginary time step to obtain a wave function for a current imaginary time step.

2. updating the wave function for the previous imaginary time step using the classical shadow of the trial wave function determining a Walker wave function for the current imaginary time step; determining Walker weights for the current imaginary time step using a first dot product of the trial wave function and a Walker wave function for the previous imaginary time step and a second dot product of the trial wave function and a Walker wave function for the current imaginary time step, wherein the first and second dot products are determined using the classical shadow of the trial wave function.

3. 3. The method of claim 1 or claim 2, further comprising storing the calculated classical shadow of the trial wave function in a classical memory of the classical computer.

4. determining a Walker weight for the current imaginary time step using the first dot product of the trial wave function and the Walker wave function for the previous imaginary time step and the second dot product of the trial wave function and the Walker wave function for the current imaginary time step, comprising: retrieving the classical shadow of the trial wave function from the classical memory; calculating an approximation of the first dot product, comprising determining expectation values ​​of one or more classically simulated first projectors and the classical shadow of the trial wave function, wherein the one or more first projectors depend on the Walker wave function for the previous imaginary time step; and calculating an approximation of the second dot product comprising determining expectation values ​​of one or more classically simulated second projectors and the classical shadow of the trial wave function, the one or more second projectors depending on the Walker wave function for the current imaginary time step.

5. The method of claim 4 , wherein the one or more first projectors are generated using a stabilizer state.

6. The method of claim 5 , wherein the stabilizer state comprises a computational basis state with a Hamming weight equal to the number of particles represented by the trial state.

7. 7. The method of claim 1, wherein the trial wave functions include trial wave functions rotated with unitary operators randomly sampled from an ensemble of unitaries, the ensemble of unitaries being tomographically complete.

8. The method of claim 7 , wherein the unitary operator comprises a tensor product of an N-qubit Clifford circuit or randomly selected Clifford circuits with fewer than N qubits.

9. 9. The method of claim 1, wherein performing each imaginary time step of the imaginary time propagation further comprises calculating an energy estimate using the classical shadow of the trial wave function.

10. The trial wave functions include trial wave functions transformed using a tensor product of unitary operators, each unitary operator in the tensor product being a respective one of the randomly selected N p∈P Including the Qubit Clifford gate, N p∈P 10. A method according to any one of claims 1 to 5 or 9, wherein p denotes the number of qubits in part p of the division of N qubits into P parts.

11. preparing, by a quantum computer, multiple copies of the trial wave function, the trial wave function approximating the target wave function; performing, by the quantum computer, measurement operations on transforms of the multiple copies of the trial wave function; and transmitting, by the quantum computer to the classical computer, data representing a result of the measurement operation.

12. 1. A computer-implemented method for performing a quantum Monte Carlo simulation of a fermionic quantum system to calculate a target wave function of the fermionic quantum system, comprising: preparing, by a quantum computer, multiple copies of a trial wave function, the trial wave function approximating the target wave function; performing, by the quantum computer, measurement operations on transforms of the multiple copies of the trial wave function; transmitting data representing the results of the measurement operation by the quantum computer to a classical computer, wherein the classical computer uses the transmitted data to perform imaginary time propagation of an initial wave function using a Hamiltonian that characterizes the fermionic quantum system.

13. performing a measurement operation on a transform of a copy of the trial wave function, randomly sampling a unitary operator from an ensemble of unitary operators, wherein the ensemble of unitary operators is tomographically complete; applying the randomly sampled unitary operator to the copy of the trial wave function to obtain a rotated trial wave function; and measuring the rotated trial wave function on the computational basis.

14. performing a measurement operation on a transform of a copy of the trial wave function, Randomly sampling a plurality of unitary operators from an ensemble of unitary operators, wherein the ensemble of unitary operators is tomographically complete and each sampled unitary operator is N p∈P Including the Qubit Clifford gate, N p∈P represents the number of qubits in part p of the partition of N qubits into P parts; and applying a tensor product of the randomly sampled unitary operators to the copy of the trial wave function to obtain a transformed trial wave function; and measuring the transformed trial wave function on the computational basis.

15. 15. The method of claim 1, wherein the quantum Monte Carlo simulation comprises a projector quantum Monte Carlo simulation or an auxiliary field quantum Monte Carlo simulation.

16. 16. The method of claim 1, wherein the quantum computer comprises a noisy intermediate-scale quantum device.

17. 17. The method of claim 1, wherein the trial wave function comprises a wave function from a generalized valence bond complete pairing wave function hypothesis.

18. 18. The method of claim 17, wherein the generalized valence bond complete pairing wave function hypothesis includes a first set of layers including density-density terms and a second set of layers including nearest-neighbor hopping terms between the same spin pairs.

19. one or more computers; and one or more computer-readable storage media coupled to the one or more computers, the instructions storing instructions that, when executed by the one or more computers, cause the one or more computers to perform operations in accordance with the method of any one of claims 1 to 11 and 15 to 18.

20. one or more quantum computers; and one or more computer-readable media coupled to the one or more quantum computers that store instructions that, when executed by the one or more quantum computers, cause the one or more quantum computers to perform operations in accordance with the method of any one of claims 12 to 18.

21. 21. The system of claim 20, wherein the quantum computer comprises a NISQ device.

Images