Implementation of Unbiased Fermi - Hubbard Monte Carlo Method Using a Quantum Computer and Shadow Tomography

Abstract

In one aspect, a method, system, and apparatus are provided for hybrid quantum-classical quantum Monte Carlo. In one aspect, the method includes receiving, by a classical computer, data generated by a quantum computer, the data representing results of a measurement of a trial wave function, the trial wave function approximating a 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 results of the measurement of the 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 that characterizes the fermionic quantum system, the imaginary time propagation being performed until a predetermined convergence criterion is met, performing each imaginary time step of the imaginary time propagation includes updating the wave function for a previous imaginary time step using the classical shadow of the trial wave function to obtain a wave function for a current imaginary time step.

Description

Technical Field

[0001] This specification relates to quantum computing.

Background Art

[0002] Calculating the exact solution of the Schrödinger equation for the ground state of a multi-electron system is applied across almost all fields of modern science and enables a detailed understanding of important unsolved questions in chemistry, physics, materials science, and biology. However, the complexity of the Schrödinger equation increases exponentially with the number of electrons in the system. Therefore, progress towards efficient means of accurately calculating the ground state quantum mechanical properties of complex systems has been slow.

[0003] Known general methods for calculating the solution of the Schrödinger equation can be grouped into two categories. The first category includes methods that scale exponentially with the system size and yield a numerically exact answer. The second category includes methods that scale polynomially with the system size at a high cost and rely on error cancellation when calculating observable quantities. The methods in the second category are the only ones that can be feasibly applied to large-scale systems at present, but the accuracy of the solutions obtained in such cases is not satisfactory and is almost always difficult to utilize.

[0004] Quantum computing provides an alternative computational paradigm that can complement and potentially outperform classical methods in terms of efficiency. In the absence of fault-tolerant quantum computers, noisy intermediate-scale quantum (NISQ) technologies can be used to investigate multi-system quantum problems. Most NISQ algorithms for calculating quantum ground states are centered around the variational quantum eigenvalue solver (VQE) framework, which requires dealing with optimization problems and noisy gradients. Alternatively, algorithms based on imaginary time evolution, which avoid optimization problems in principle, have been proposed. However, due to the non-unitarity of the imaginary time evolution method, optimization heuristics must be used to achieve reasonable scaling with system size. Therefore, alternative computational strategies are needed to avoid these limiting factors in order to enable the first practical quantum supremacy in fermionic simulations. Summary of the Invention Means for Solving the Problems

[0005] This specification describes a quantum-classical hybrid algorithm for implementing an unbiased fermionic quantum Monte Carlo method using a quantum computer and shadow tomography.

[0006] Generally, one inventive aspect of the subject matter described in this specification can be implemented as a method for performing a quantum Monte Carlo simulation of a fermionic quantum system to compute a target wave function of the fermionic quantum system. The method includes steps of receiving, by a classical computer, data generated by a 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 being prepared by the quantum computer; computing, by the classical computer, a classical shadow of the trial wave function using the data representing results of 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 is performed until a predetermined convergence criterion is met. Performing each imaginary time step of the imaginary time propagation includes updating a wave function for a previous imaginary time step using the classical shadow of the trial wave function to obtain a wave function for the current imaginary time step.

[0007] Other implementations of these aspects include corresponding computer systems, each configured to perform the actions of the method, and a computer program recorded on one or more computer storage devices. A system consisting of one or more classical and / or quantum computers can be configured to perform particular operations or actions in operation, thanks to software, firmware, hardware, or combinations thereof installed on the system to cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions when executed by a data processing apparatus, thanks to instructions included in the computer programs to cause the apparatus to perform the actions.

[0008] The above and other implementations may each optionally include one or more of the following features, either alone or in combination. In some implementations, the step of updating the wave function for the previous imaginary time step using the classical shadow of the trial wave function includes the step of determining the walker wave function for the current imaginary time step, and using the first inner product of the trial wave function and the walker wave function for the previous imaginary time step and the second inner product of the trial wave function and the walker wave function for the current imaginary time step to determine the walker weight for the current imaginary time step, where the first inner product and the second inner product are determined using the classical shadow of the trial wave function.

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

[0010] In some implementations, the step of determining the walker weight for the current imaginary time step using the first inner product of the trial wave function and the walker wave function for the previous imaginary time step and the second inner product of the trial wave function and the walker wave function for the current imaginary time step includes the step of retrieving the classical shadow of the trial wave function from classical memory, and calculating an approximation of the first inner product including determining the expected value of one or more classically simulated first projectors and the classical shadow 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 calculating an approximation of the second inner product including determining the expected value of one or more classically simulated second projectors and the classical shadow 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, one or more of the first projectors are generated using a stabilizer state.

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

[0013] In some implementations, the transformed trial wave function includes a trial wave function rotated using a unitary operator randomly sampled from a unitary ensemble, where the unitary ensemble is tomographically complete. In some implementations, the unitary operator includes the tensor product of an N-qubit Clifford circuit or a randomly selected Clifford circuit on less than N qubits.

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

[0015] In some implementations, the transformed trial wave function includes a trial wave function transformed using the tensor product of unitary operators, where each unitary operator in the tensor product is a randomly selected N p∈P qubit Clifford gate, and N p∈P represents the number of qubits in the portion p of the partition of the N qubits into P parts.

[0016] In some implementations, the quantum Monte Carlo simulation includes projective quantum Monte Carlo simulation or 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 complete pairing wave function hypothesis.

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

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

[0021] A system implementing the techniques described in this application can target quantum states and their properties with improved computational efficiency and improved accuracy. For example, in this hybrid quantum-classical quantum Monte Carlo algorithm, the classically implemented quantum Monte Carlo method does not need to repeatedly query a quantum computer. By separating the interaction between the quantum and classical computers in this way, the need to minimize latency is avoided, which is a particularly attractive feature on NISQ platforms.

[0022] Furthermore, a system implementing the techniques described in this application uses a trial wave function that is inherently more accurate than conventional trial wave functions, such as single determinants, and can be obtained by an efficient polynomial scaling classical technique that circumvents the problems of variational optimization in quantum computers. The trial wave function can include wave functions for which there are no known polynomial scaling classical algorithms for the evaluation of the quantities required by the quantum Monte Carlo method. The trial wave function achieves polynomial scaling, and thus the techniques described in this application achieve an exponential speedup in computational speed compared to classical ones.

[0023] Furthermore, a system implementing the techniques described in this application can calculate quantities required by the quantum Monte Carlo method, such as wave function overlap, with a bounded number of experiments and measurement repetitions (without constraints on the form of the trial wave function). This number is of order

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[0024] Furthermore, the techniques described in this application are robust against noise, for example, noise resulting from hardware defects, because the quantity directly calculated is the ratio between overlapping values, which is inherently strong against overlaps that are rescaled by several error channels.

[0025] 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

[0026]

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Modes for Carrying Out the Invention

[0027] Quantum Monte Carlo (QMC) methods use the many-body Hamiltonian

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[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,

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[0029] Formally exact, such stochastic imaginary-time evolution algorithms generally encounter the well-known fermion sign problem, which manifests itself through the alternating signs in the weights of each statistical sample. In the worst case, the fermion sign problem causes the energy estimator to have an exponentially large variance, and to obtain a fixed-precision estimate of observable quantities such as the ground-state energy, it is necessary to average an exponentially large number of samples. Therefore, reliable calculations of the ground state and its properties are practically impossible to achieve, and exact unbiased QMC methods are applicable only to small-scale systems or those without the sign problem.

[0030] In the first quantization QMC method, this problem manifests as a bosonic ground state. Since fermion antisymmetry is not explicitly imposed, the true ground state of the first quantization Hamiltonian is actually bosonic. This then requires imposing a fermionic nodal structure in the first quantization to calculate the fermionic ground state. In the second quantization QMC method, bosonic states cannot be obtained from the fermionic Hamiltonian. The sign problem appears differently. Statistical estimators from the second quantization QMC method exhibit a variance that increases exponentially with the system size.

[0031] The sign problem can be controlled to give an estimator of the ground-state energy with polynomially bounded variance by imposing constraints on the imaginary-time evolution method for each statistical sample, which is represented by each wave function, i.e., |φ i (τ)〉. These constraints, such as the fixed-node and phase-free approximations, can be imposed by the use of a trial wave function |Ψ T 〉, and the accuracy of constrained QMC is determined by the choice of the trial wave function. Such constraints necessarily introduce a potentially large bias into the final ground-state energy estimate.

[0032] Classically, computationally tractable options for the trial wave function 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 (generally called a Jastrow factor) applied to the mean-field state, or some other physically motivated transformation applied to the mean-field state such as the backflow approach. On the other hand, wave functions that can be prepared with a quantum circuit are candidates for the trial wave function on a quantum computer, including more general two-body correlators. These trial wave functions are referred to herein as “quantum” trial wave functions.

[0033] This specification describes a hybrid quantum-classical QMC algorithm that combines constrained quantum Monte Carlo (QMC) with quantum computing techniques to reduce bias in the estimated final quantum state. The quantum-classical hybrid QMC algorithm (QC-QMC) uses a quantum trial wave function while performing most of the imaginary-time evolution method on a classical computer. That is, the classical computer performs the imaginary-time evolution method for each statistical sample |φ i (τ)〉 and collects observable quantities such as the estimated ground-state energy E (i) (τ). During this procedure, constraints by the quantum trial wave function are imposed to control the sign problem.

[0034] To perform 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 i(τ)> is a calculation of the overlap with. In particular, the QC-QMC algorithm described in the present application uses shadow tomography to estimate the overlap between the trial wave function and the statistical sample. Based on experiments, this involves performing a randomly selected set of measurements of the reference state associated with the trial wave function prior to starting the QMC method. This enables an efficient estimation of the entire set of required overlaps using a small number of experimental repetitions 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, regardless of the fact that the details of the statistical sample are not determined in advance. By separating the interaction between the quantum and classical computers, the need to minimize latency is avoided, which is a particularly attractive feature on NISQ platforms.

[0035] The QC-QMC algorithm described in the present application generally applies to any form of constrained QMC. For illustrative purposes, this specification describes a specific demonstration of the QC-QMC algorithm that uses an implementation form of QMC known as auxiliary-field QMC (AFQMC). AFQMC is a successful PQMC method in the second quantization space. Therefore, the sign problem in AFQMC appears when increasing the variance in the statistical estimate. To impose constraints in imaginary-time propagation, importance sampling and trial wave functions that can be used in the constraints are incorporated. As a result, the wave function at imaginary time τ is

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

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[0037] While iteratively applying the imaginary - time propagation to the wave function, the AFQMC algorithm defines a specific technique for updating the walker weight w

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[0038] At present, classically tractable trial wave functions are either a single determinant or a linear combination of determinants. The former is scalable (up to about 500 electrons) but can often be inaccurate, especially for strongly correlated systems. The latter is limited to a small number of electrons (about 14) but can be made very accurate even for strongly correlated systems. The choice of trial wave function in AFQMC is restricted by the evaluation of Eqs. 4 and 6. The calculation for either of these scales exponentially with the system size, and as a result, the resulting AFQMC method is exponentially expensive.

[0039] The QC-QMC algorithm described in this application uses a class of trial wave functions that can be obtained by an efficient polynomial-scaling classical method, which is an essentially more accurate trial wave function than a single determinant and circumvents the problems of variational optimization in quantum computers. The trial wave functions can include wave functions for which there are no known polynomial-scaling classical algorithms for the evaluations of Equations 4 and 6. A quantum computer is used to remove such limitations by introducing polynomial-scaling algorithms for Equations 4 and 6, which, by doing so, guarantees an exponential speedup compared to classical computers. In the QC-QMC algorithm described in this application, Equations 4 and 6 can be measured on a quantum computer, and the actual imaginary-time propagation can be implemented classically. This divides the subroutines into those that need to run on a quantum computer and those that need to run on a classical computer.

[0040] In some implementations, the trial wave function can be a variant of the coupled cluster wave function. The coupled cluster wave function is characterized by exponential parameterization, i.e.,

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[0041] Using CCSD (or other higher-order coupled cluster wavefunctions) is not suitable for use as an AFQMC trial wavefunction because the projection of such functions onto any Slater determinant cannot be efficiently calculated without approximation. This applies to almost all non-trivial variants of coupled cluster. The cost of calculating the wavefunction overlap for the coupled cluster method with a limited set of amplitudes such as generalized valence bond perfect pairing (PP) 1,2 also scales exponentially with the system size. The required overlap of such wavefunctions can be efficiently evaluated by using a quantum computer to prepare the unitary version of the coupled cluster wavefunctions or approximations thereto. By using coupled cluster wavefunctions that can be classically optimized, the costly variational optimization procedure in quantum devices is avoided.

[0042] An exemplary coupled cluster wavefunction ansatz that can be used as a trial wavefunction is the generalized valence bond PP ansatz. This ansatz is

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[0043] PP wave functions are often insufficient to achieve qualitative accuracy. This is most evident in systems where pair correlations, such as multiple bond breaking, become important. There are several ways to classically incorporate those pair correlations, but the QC-QMC multiple layers of hardware-efficient operators described in this application 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

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

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[0045] Multiple layers of various types are alternated and applied to the PP hypothesis to improve overall accuracy. The effectiveness of these layers varies with the choice of the i, j pair.

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

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

[0048] Quantum processor 102 includes components for performing quantum computing. For example, quantum processor 102 may include a qubit array, a quantum circuit configuration, and a control device configured to operate the physical qubits in the qubit array and apply a quantum circuit to the qubits. Exemplary quantum processors are described in more detail below with reference to FIG. 5.

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

[0050] As described above, the QC-QMC algorithm described in the present 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 a quantum state without relying on full state tomography. Let ρ denote some unknown quantum state. It is assumed that access to N copies of ρ is possible. Let {O i} denote a collection of M observables. The task is to estimate the quantity Tr(ρO i ) up to some additive error ε for each O i . This can be efficiently accomplished 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 the protocol, a unitary ensemble u is chosen. Then, a unitary U k ∈ u is randomly sampled and the state

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[0052] M is required to be invertible, which holds only if the set of measurement operators defined by drawing 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, [Mathematics] results in. Aggregate [Mathematics] is the classical shadow of ρ. Many choices are possible for the ensemble u. For example, a randomly selected N - qubit Clifford circuit, as well as the tensor product of randomly selected Clifford circuits for fewer qubits, can be used.

[0053] Therefore, in stage (A) of the QC - QMC algorithm, the quantum processor 102 performs a randomly selected set of measurements of copies of the trial wave function for the QMC method. That is, the quantum processor 102 measures the quantum state [Mathematics] and performs a plurality of experiments to collect the corresponding measurement data. In stage (B) of the QC - QMC algorithm, the quantum processor 102 sends the collected measurement data to the classical processor 104, and by doing so, the classical processor 104 can implement the QMC algorithm. Stages (A) and (B) can be performed prior to the QMC algorithm.

[0054] For each of the plurality of experiments, the quantum processor 102 can apply a quantum circuit to the physical qubits included in the quantum processor 102. The circuit can include a first circuit that prepares the qubits in an initial state, for example, a superposition of the trial wave function and the zero state, and a second quantum circuit that implements the measurement operator for the shadow tomography experiment. The specific form of the first and second circuits depends on the trial wave function used.

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

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

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[0057] In some implementations, the following global stabilizer measurement strategy can be implemented to reduce the size of the quantum circuit required to perform shadow tomography. Generally, applying the unitary U and then measuring in the computational basis {|x〉: x ∈ {0, 1} n} as shadow tomography was originally presented is equivalent to measuring in the rotated basis {U † |x〉: x ∈ {0, 1} n}. For a set of unitaries u, randomly and uniformly choosing a unitary from there and then measuring in the computational basis is equivalent to measuring a POVM

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[0058] F n is the "H - free" group for n qubits, that is, the group generated by X, CNOT, CZ. The action of any H - free operator can

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[0059] Furthermore, in some implementations, a segmented shadow tomography strategy can be implemented to reduce the quantum circuit depth. This strategy is described in detail below with reference to FIG. 2.

[0060] In stage (C) of the QC - QMC algorithm, the classical processor processes the received measurement results and computes a classical shadow. The classical shadow can be stored in the classical memory 106 of the classical processor 104.

[0061] In stage (D) of the QC - QMC algorithm, the classical processor 104 uses the stored classical shadow to implement the QMC algorithm. That is, the classical processor 104 performs imaginary time propagation, for example, for a sequence of imaginary time steps of an initial wave function using a Hamiltonian that characterizes a fermionic quantum system according to Equation 1. In each imaginary time step, the classical processor uses the stored classical shadow to compute the required wave function overlaps. Exemplary operations performed by the 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 the 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, such as the expected energy of the target quantum state, as described above with respect to equations 2 and 4-6.

[0063] FIG. 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 of the fermionic quantum system and / or a property of the target wave function, such as a ground state energy. For convenience, process 200 is described as being performed by a system that includes classical and quantum computing devices at one or more locations. For example, system 100 of FIG. 1, appropriately 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 a generalized valence bond complete pairing wave function ansatz, which includes a first set of layers including density-density product terms and a second set of layers including nearest neighbor hopping terms between the same spin pairs.

[0065] The system uses a quantum computing device to perform measurement operations 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 using unitary operators randomly sampled from a unitary ensemble, where the unitary ensemble is tomographically complete. The unitary operators used may be the tensor product of an N-qubit Clifford circuit or a randomly selected Clifford circuit for less than N qubits. For example, the transformed trial wave function can be given as described above in the discussion around Equation 14. [Number] The quantum computing device can then measure the rotated trial wave function in the computational basis to obtain each measurement result. This process can be repeated for each copy of the traveling wave function.

[0066] In some implementations, the system can partition the 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 the tensor product of unitary operators to the trial wave function, where each unitary operator in the tensor product is a randomly selected N p∈P qubit Clifford gate, and N p∈P represents the number of qubits in partition p of the partition of the N qubits into P parts as described below with respect to Equations 29 - 34. The quantum computing device can then measure the transformed trial wave function in the computational basis to obtain each measurement result.

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

[0068] The classical computing device receives data representing the results of measurements of the transformed trial wave functions generated by the quantum computing device and uses this data to compute the classical shadow of the trial wave functions (step 208). The computation of the classical shadow was described above with respect to equations 13 and 14. The classical computing device can efficiently store the computed classical shadow in the classical memory of the classical computing device.

[0069] The system uses a classical computing device to perform imaginary time propagation of the initial wave function (for a sequence of imaginary time steps) using the Hamiltonian characterizing the fermionic quantum system (step 210). The imaginary time propagation may be performed until a predetermined convergence criterion is met, e.g., until the output wave function converges 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 uses the classical shadow of the trial wave function to update the wave function for the previous imaginary time step 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 the walker wave function for the current time step, e.g., through imaginary time propagation, and determines the walker weight for the current time step using i) a first inner product of the trial wave function and the walker wave function for the previous time step and ii) a second inner product of the trial wave function and the walker wave function for the current time step, where the first and second inner products are determined using the classical shadow of the trial wave function.

[0071] Exemplary techniques performed by the system to determine the inner product of the trial wave function and the walker wave function using the classical shadow include the following. |Ψ T〉 shall represent a trial wave function. In some implementations, |Ψ T 〉 can be chosen to represent a fermionic wave function with a fixed number of particles η > 0, and a quantum state encoded by the Jordan-Wigner transformation can be used. By doing so, the qubit wave function for |Ψ T 〉 becomes a superposition of computational basis states and Hamming weight η.

[0072] Let |φ〉 represent a Walker wave function. The Walker wave function can be a superposition of computational basis states and Hamming weight η. Calculating the inner product of the trial wave function and the Walker wave function may thus involve calculating the inner product 〈φ|Ψ T 〉 using the classical shadow of the trial wave function.

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

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[0074] When the ensemble u used to generate the classical shadow is the Clifford group for N qubits, the inverse of the channel M can be given by

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[0075] Overlap between stabilizer states (including the ground state) can be classically computed efficiently using the Gottesman-Knill theorem, and the right-hand side of Equation 28 can be classically computed efficiently. In particular, $\langle b k |U k |0\rangle$ can be efficiently computed for any Clifford circuit $U k $, and since the Walker wavefunction can be written as a linear combination of a polynomial number of stabilizer states, the quantity

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[0076] As described above, shadow tomography using the $N$-qubit Clifford group can be used to simultaneously estimate $M$ quantities, as in Equation 27, with a cost that scales logarithmically with $M$. However, performing these measurements on a NISQ device can be difficult due to the required circuit depth. An alternative preference for the ensemble of random unitaries $u$ can mitigate this issue. A second preference for $u$ includes a unitary $U\in u$ 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 Cliffords for $U$ leads to a limit on the cost of shadow tomography that scales exponentially with the locality of the operator being estimated. Projectors are highly non-local operators, and thus one might expect to encounter a large number of measurement repetitions required when using single-qubit Clifford shadow tomography to estimate their expectation values (assuming that practical implementation is correlated with the limit). This suggests that a trade-off between circuit depth and the number of repetitions required to perform shadow tomography with different preferences for $U$ should be considered.

[0077] For that purpose, using $u$ consisting of randomly sampled tensor products of Clifford unitaries for qubits fewer than $N$, $\langle\beta|\PsiT Alternative techniques can be implemented to efficiently perform the classical post - processing required to estimate 〉. The expression in Equation 28 can also be written as follows.

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[0078] |0〉 = 0. For this, the inner product is p |0〉 = 0. For this, the inner product is [Number] can be classically calculated as, in the above formula, c i represents the amplitude of |φ〉 in the computational basis {|β i 〉}.

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

[0080] At each imaginary time step of the imaginary time propagation, the classical computer also calculates, for example, the energy estimator given by Equation 3 using the classical shadow of the trial wave function. In some implementations, the ground state energy is estimated from the time series of energy estimators calculated at each imaginary time step.

[0081] Figure 3 shows the application of the QC-QMC algorithm described in this application to the H4 molecule in an 8-qubit experiment. In this example, an 8-spin orbital quantum trial wave function is used. The trial wave function consists of a valence bond wave function known as the full pairing state, to which a hardware-efficient quantum circuit with offline single-particle rotations is applied. Classically, this would be difficult to use as a trial wave function for AFQMC.

[0082] Part (a) of Figure 3 shows an exemplary state preparation circuit for preparing the trial wave function using a quantum computer. In this 8-qubit experiment, H4 within a square with a side length of 1.23 Å and dissociated into 4 hydrogen atoms is considered. This system can be used as a test bench for electron correlation methods in quantum chemistry. Part (a) shows the experimental circuit used for the experiment over a 2×4 qubit grid. In the circuit diagram, H represents the Hadamard gate, G represents the Gibbs rotation gate (generated by the Pauli gate (XX + YY)), P represents the Pauli gate, and |Ψ T 〉 represents the quantum trial wave function. The offline orbital rotation does not exist in the actual quantum circuit since it can be efficiently handled by classical post-processing.

[0083] Parts (b) and (c) of FIG. 3 show the convergence of the atomization energy of H4 as a function of the number of measurements. Part (b) shows the minimal basis set (STO-3G) with four orbitals from a total of four separate experiments with different sets of random measurements, and part (c) shows the quadruple zeta basis set (cc-pVQZ) with 120 orbitals from a total of two separate experiments. The different symbols in (b) and (c) represent separate experimental results. The upper panels of (b) and (c) expand the energy range near the exact answer. As shown, due to the noise in the quantum device, the quality of the quantum trial deviates from that of the ideal (i.e., noiseless) hypothesis, resulting in an error of up to 10 kcal / mol in the atomization energy. Nevertheless, the QC-AFQMC described in this application significantly reduces this error and achieves chemical accuracy for both bases. To further clarify the QC-AFQMC results for H4, parts (b) and (c) show the evolution of the trial and the QC-AFQMC energy as a function of the number of measurements performed on the device. Despite the presence of significant noise at approximately 105 measurements, QC-AFQMC achieves chemical accuracy while addressing the substantial residual bias in the underlying quantum trial.

[0084] FIG. 4 is a flowchart of an exemplary process 400 for performing a quantum Monte Carlo simulation of a fermionic quantum system to calculate a target wave function of the fermionic quantum system and / or properties of the target wave function, such as a ground wave function energy. In some implementations, the quantum Monte Carlo simulation may be a projected quantum Monte Carlo simulation, such as an auxiliary field quantum Monte Carlo simulation. For convenience, process 400 is described as being implemented by a system that includes classical and quantum computing devices at one or more locations. For example, system 100 of FIG. 1 appropriately programmed in accordance with this specification can implement process 400.

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

[0086] Each imaginary-time step of the 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, for example, a NISQ device (step 404). The quantum computer calculates an inner product using data representing the wave function for the previous wave function and a trial wave function that approximates the target wave function. Exemplary trial wave functions were described above with reference to FIG. 1.

[0087] The classical computer receives data representing the calculated inner product, generated by the quantum computer (step 408), and uses the data representing the calculated inner 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 can also calculate an energy estimator using the classical shadow of the trial wave function, for example, by calculating Equation 3.

[0088] In some implementations, the classical computer uses the data representing the calculated inner product to determine the walker wave function for the current time step and to determine the walker weight for the current time step, thereby updating the wave function for the previous imaginary time step to obtain the wave function for the current imaginary time step, where the calculated inner product includes the first inner product of the trial wave function and the walker wave function for the current time step and the second inner product of the trial wave function and the walker wave function for the previous time step. That is, the classical computer uses Equations 3-6 to update the wave function for the previous imaginary time step, where the inner product is calculated by the quantum computer. In these implementations, the data representing the wave function for the previous imaginary time step transmitted from the classical computer to the quantum computer includes the data representing the walker wave function for the previous imaginary time step and the data representing the calculated walker wave function for the current imaginary time step (e.g., calculated by the classical computer by imaginary time propagation).

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

[0090] The quantum computer can calculate the inner product using a projective measurement on the trial wave function, where the projection operator of the projective measurement is generated using a stabilizer state. The stabilizer state can include a computational basis state having a Hamming weight equal to the number of particles represented by the trial wave function. The projection operator of the projective measurement can be determined by the data representing the walker wave function for the previous imaginary time step or the data representing the calculated walker wave function for the current imaginary time step. The calculation of the inner product and the projective measurement that can be performed by the quantum computer are described above with reference.

[0091] Figure 5 shows an exemplary classical / quantum computer 500 for performing 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 herein, their connections and relationships, and their functions are exemplary only and do not limit the implementations of the invention described and / or claimed herein.

[0092] The 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, such as qubit 506, used to perform algorithm operations or quantum computations. The qubits shown in Figure 5 are arranged in a rectangular array, but this is a schematic description and is not intended to be limiting. The qubit assembly 552 also includes adjustable coupling elements, such as coupler 508, that enable interaction between the coupled qubits. In the schematic description of Figure 5, each qubit is adjustably coupled to each of its four adjacent qubits using respective coupling elements. However, this is an exemplary arrangement of qubits and couplers, and other arrangements are possible, including non-rectangular arrangements, arrangements that enable 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 having levels representing logical values of 0 and 1. The specific physical realizations of multiple qubits and how they interact with each other depend on various factors, including the type of quantum computing device 502 included in the exemplary computer 500 or the type of quantum computing being performed by the quantum computing device. For example, in an atomic quantum computer, qubits may be realized by atoms, molecules, or solid-state quantum systems, such as ultrafine atomic states. As another example, in a superconducting quantum computer, qubits may be realized by superconducting qubits or semiconductor qubits, such as superconducting transmon states. As another example, in an NMR quantum computer, qubits may be realized by nuclear spin states.

[0094] In some implementations, quantum computing may proceed, for example, by loading qubits from a quantum memory and applying a sequence of unitary operators to the qubits. Applying a unitary operator to a qubit may include, for example, applying a corresponding sequence of quantum logic gates to the qubit to implement the quantum circuit required for shadow tomography, as described above with reference to FIG. 1. Exemplary quantum logic gates include single-qubit gates, such as Pauli X, Pauli Y, Pauli Z (also called X, Y, Z), Hadamard gate, S gate, rotation, two-qubit gates, such as controlled X, controlled Y, controlled Z (also called CX, CY, CZ), controlled NOT gate (also called CNOT), controlled swap gate (also called CSWAP), iSWAP gate, and gates involving three or more qubits, such as the Toffoli gate. 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 the qubit assembly 552 can have variable frequencies. In these examples, each qubit can have an associated operating frequency that can be adjusted by applying a voltage pulse via one or more drive lines coupled to the qubit. Exemplary operating frequencies can include the qubit idle frequency, the qubit interaction frequency, and the qubit readout frequency. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to the corresponding idle frequency may put the qubit in a state where it does not strongly interact with other qubits and can be used to perform single qubit gates. As another example, in cases where qubits interact via a coupler with fixed couplings, the qubits may be configured to interact with each other by setting their respective operating frequencies to some gate-dependent frequency detuning from a common interaction frequency. In other cases, for example, when qubits interact via a tunable coupler, the qubits can be configured to interact with each other by first setting the parameters of their respective couplers to enable interaction between the qubits, and then setting the respective operating frequencies of the qubits to some gate-dependent frequency detuning from their common interaction frequency. Such interactions may be performed 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 can include RF or microwave pulses in an NMR or superconducting quantum computer system, or optical pulses in an atomic quantum computer system.

[0097] By measuring the state of the qubits using respective control signals 510, quantum computing can be completed, for example, using quantum observables such as X or Z. Through the measurement, a readout signal 512 representing the measurement result is communicated back to the measurement and control system 504. The readout signal 512 may include RF, microwave, or optical signals depending on the physical scheme for the quantum computing device and / or qubits. For the sake of convenience, although the control signal 510 and the readout signal 512 shown in FIG. 5 are shown as addressing only selected elements (i.e., the upper and lower rows) of the qubit assembly, during operation, the control signal 510 and the readout signal 512 can address each element within the qubit assembly 552.

[0098] The control and measurement system 504 is an example of a classical computer system that can be used to perform various operations on the qubit assembly 552 as described above, as well as other classical subroutines or calculations. The control and measurement system 504 includes one or more classical processors, such as the classical processor 514, one or more memories, such as the memory 516, and one or more I / O units, such as the I / O unit 518, connected by one or more data buses. The control and measurement system 504 may be programmed to send a sequence of control signals 510 to the qubit assembly, for example, to perform a selected series of quantum gate operations, and to receive a sequence of readout signals 512 from the qubit assembly, for example, 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] Memory 516 stores information within control and measurement system 504. In some implementations, memory 516 includes a computer-readable medium, a volatile memory unit, and / or a non-volatile memory unit. In some cases, memory 516 can include a storage device capable of providing mass storage to system 504, such as a hard disk device, an optical disk device, a storage device shared by multiple computing devices via a network (e.g., a cloud storage device), and / or any other mass storage device.

[0101] Input / output device 518 provides input / output operations for control and measurement system 504. Input / output device 518 can include a D / A converter, an A / D converter, and an RF / microwave / optical signal generator, transmitter, and receiver for sending control signal 510 to and receiving readout signal 512 from qubit assembly, as required by the physical approach for a quantum computer. In some implementations, input / output device 518 can also include one or more network interface devices, such as an Ethernet card, a serial communication device, such as an RS-232 port, and / or a wireless interface device, such as an 802.11 card. In some implementations, input / output device 518 can include a driver device configured to receive input data and send output data to other external devices, such as a keyboard, a printer, and a display device.

[0102] FIG. 5 shows an exemplary control and measurement system 504, but implementations of the subject matter and functional operations described in this specification can be implemented in other types of digital electronic circuitry, or in computer software, firmware, or hardware that includes the structures disclosed in this specification and their structural equivalents, or in a combination of one or more of them.

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

[0104] Implementations of the subject matter and the operations described in this specification can be realized in digital electronic circuitry, analog electronic circuitry, appropriate quantum circuitry, or, more generally, in a quantum computing system, in software or firmware tangibly embodied in a computer hardware including the structures disclosed in this specification and their structural equivalents, or in a combination of one or more of them. The term "quantum computing system" can 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 in this specification 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 to control 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 them. 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 suitable receiver device for execution by a data processing apparatus.

[0106] The terms "quantum information" and "quantum data" refer to information or data that is carried by and held or stored in a quantum system, where the smallest non-trivial system is the qubit, i.e., the system that defines the 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 can include, for example, multi-level systems having 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, but it will be understood that other setups are possible where the computational states are identified with higher-level excited states.

[0107] The term "data processing apparatus" refers to digital and / or quantum data processing hardware and encompasses all kinds of devices, apparatuses, and machines for processing digital and / or quantum data, including, by way of example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The apparatus may further include, or alternatively be, special-purpose logic circuit configurations, such as FPGAs (field-programmable gate arrays), ASICs (application-specific integrated circuits), or quantum simulators, i.e., quantum data processing apparatuses designed to simulate or give rise to information about specific quantum systems, or even further include them. In particular, a quantum simulator is a special-purpose quantum computer that does not have the ability to perform universal quantum computing. The apparatus may optionally include, in addition to the hardware, code that creates an execution environment for digital and / or quantum computer programs, such as processor firmware, protocol stacks, database management systems, operating systems, or code that constitutes one or more combinations thereof.

[0108] A digital computer program, which may be called or known as a program, software, software application, module, software module, script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and 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 be called or known as a program, software, software application, module, software module, script, or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, may be translated into a suitable quantum programming language, or may be written in a quantum programming language, such as QCL or Quipper.

[0109] A computer program may, but need not, correspond to a file in a file system. The program may be stored in another program or data, such as a file portion that holds 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, such as files that store one or more modules, subprograms, or portions of code. A computer program may be deployed to execute on one computer located in one place or on multiple computers, or may be distributed across multiple locations and interconnected by digital and / or quantum data communication networks. A quantum data communication network is understood to be a network that can transmit quantum data using a quantum system, such as a qubit. Generally, a digital data communication network cannot transmit quantum data, but a quantum data communication network can transmit both quantum data and digital data.

[0110] The processes and logical flows described herein can be implemented, where appropriate, by one or more programmable computers executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logical flows can also be implemented by special purpose logic circuitry, such as an FPGA or ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or a quantum simulator and one or more programmed digital and / or quantum computers, and the apparatus can be implemented as such.

[0111] A system consisting 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 causes the system to perform the operation or action during operation. One or more computer programs being configured to perform a particular operation or action means that the one or more programs, when executed by a data processing apparatus, include instructions that cause the apparatus to perform the operation or action. For example, a quantum computer may receive instructions from a digital computer that cause the quantum computing apparatus to perform an operation or action when executed by the quantum computing apparatus.

[0112] A computer suitable for the execution of a computer program can be based on a general-purpose microprocessor or a dedicated processor, or other types of central processing units. Generally, 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, such as photons, or a combination thereof.

[0113] 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 or incorporated into a dedicated logic circuit or a quantum simulator. Generally, a computer also includes one or more mass storage devices for storing data, such as magnetic, magneto-optical disks, optical disks, or a quantum system suitable for storing quantum information, or is operatively coupled to a mass storage device to receive data from, transfer data to, or both from and to the mass storage device. However, a computer does not necessarily have such a device.

[0114] A quantum circuit element (also called a quantum computing circuit element) includes circuit elements for performing quantum processing operations. That is, a quantum circuit element is 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 represent information that is in multiple states simultaneously and can be configured to operate on the information. Examples of superconducting quantum circuit elements include, in particular, 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).

[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, logical, 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 circuit configurations, 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 manufacturing the circuit elements described herein may involve the removal of one or more materials from the device during assembly. Depending on the materials 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 lithography techniques (e.g., photolithography or electron beam lithography).

[0117] During operation of a quantum computing system using superconducting quantum circuit elements and / or superconducting classical circuit elements such as the circuit elements described herein, the superconducting circuit elements are cooled within a cryostat to a temperature at which the superconducting material can exhibit superconducting properties. A superconducting (alternatively superconducting) material can be understood as a material that exhibits superconducting properties below the 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 base planes are formed from materials that exhibit superconducting properties below the 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 such as EPROM, EEPROM, and flash memory devices; magnetic disks such as internal hard disks or removable disks; magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems such as trapped atoms or electrons. Quantum memory is understood to be a device, such as an optical-matter interface where light is used for transmission and matter is used to store and preserve quantum features of quantum data such as superposition or quantum coherence, that can store quantum data for long periods of time with high fidelity and efficiency.

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

[0121] This specification includes many specific implementation details, but these should not be construed as limitations on the scope of what can be claimed, but rather as descriptions of features that may be specific to certain implementation forms. Also, some of the features described herein in the context of separate implementation forms can be implemented in combination in a single implementation form. Conversely, the various features described in the context of a single implementation form can also be implemented separately in multiple implementation forms, or in any suitable sub-combination. Furthermore, features are described above as acting in some combinations and may initially be claimed as such, but one or more features from the claimed combination can, in some cases, be deleted from that combination, and the claimed combination can be directed to a sub-combination or a variant of a sub-combination.

[0122] Similarly, operations are shown in the drawings in a particular order, but this should not be understood as requiring that such operations be performed in the particular order shown or in a sequential order to achieve the desired result, or that all of the illustrated operations be performed. In some situations, multitasking and parallel processing may be advantageous. Moreover, the separation of the various system modules and components in the implementation forms described above should not be understood as requiring such separation in all implementation forms, and it should be understood that the described program components and systems can generally be integrated together into a single software product or packaged into multiple software products.

[0123] Specific implementation forms of the subject matter have been described. Other implementation forms 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 the desired result. As an example, the process shown in the accompanying drawings does not necessarily require the particular order or sequence shown to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous.

Description of Reference Numerals

[0124] 100 System 102 Quantum Processor 104 Classical Processor 106 Classical Memory 500 Classical / Quantum Computer, Computer, System 502 Quantum Computing Device 504 Control and Measurement System, System 508 Coupler 514 Classical Processor, Processor 516 Memory 518 I / O Unit, Input / Output Device 550 Classical Processor 552 Qubit Assembly

Claims

**Claim 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 results of one or more measurements of a trial wave function, the trial wave function approximating the target wave function and being 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 characterizing the fermionic quantum system; wherein the imaginary-time propagation is performed until a predetermined convergence criterion is met; each step of performing an imaginary-time step of the imaginary-time propagation includes updating a wave function for a previous imaginary-time step using the classical shadow of the trial wave function to obtain a wave function for a current imaginary-time step. **Claim 2** The step of 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; determining a walker weight for the current imaginary-time step using a first inner product of the trial wave function and the walker wave function for the previous imaginary-time step and a second inner product of the trial wave function and the walker wave function for the current imaginary-time step, the first inner product and the second inner product being determined using the classical shadow of the trial wave function, the method according to claim 1. **Claim 3** The method according to claim 2, further comprising storing the calculated classical shadow of the trial wave function in classical memory of the classical computer. **Claim 4** A step of determining a walker weight for a current imaginary time step using the first inner product of the trial wave function and the walker wave function for the previous imaginary time step and the second inner product of the trial wave function and the walker wave function for the current imaginary time step is a step of retrieving the classical shadow of the trial wave function from the classical memory; a step of calculating an approximation of the first inner product, including determining an expected value 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; a step of calculating an approximation of the second inner product, including determining an expected value of one or more classically simulated second projectors and the classical shadow of the trial wave function, wherein the one or more second projectors depend on the walker wave function for the current imaginary time step, the method according to claim 3 comprising.

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

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

7. The method according to claim 1, wherein the trial wave function includes a trial wave function rotated using a unitary operator randomly sampled from a unitary ensemble, and the unitary ensemble is tomographically complete.

8. The method according to claim 7, wherein the unitary operator includes a tensor product of an N - qubit Clifford circuit or a randomly selected Clifford circuit in less than N qubits.

9. The method according to claim 1, wherein the step of performing each imaginary time step of the imaginary time propagation further includes a step of calculating an energy estimator using the classical shadow of the trial wave function.

10. The trial wave function includes a trial wave function transformed using a tensor product of unitary operators, and each unitary operator in the tensor product is a randomly selected N p∈P qubit Clifford gate, where N p∈P represents the number of qubits in sub-part p of the partition of the N qubits into P parts, according to the method of claim 1.

11. a step of preparing a plurality of copies of the trial wave function by a quantum computer, wherein the trial wave function approximates the target wave function; Performing a measurement operation on the transformation of the plurality of copies of the trial wave function by the quantum computer; The method according to claim 1, further comprising the step of transmitting, by the quantum computer, data representing the result of the measurement operation to the classical computer.

12. A computer-implemented method for performing 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, a plurality of copies of a trial wave function, wherein the trial wave function approximates the target wave function; Performing, by the quantum computer, a measurement operation on the transformation of the plurality of copies of the trial wave function; Transmitting, by the quantum computer, data representing the result of the measurement operation to a classical computer, wherein the classical computer performs imaginary-time propagation of an initial wave function using a Hamiltonian that characterizes the fermionic quantum system using the transmitted data; The imaginary-time propagation is performed until a predetermined convergence criterion is met; Performing the imaginary-time propagation of the initial wave function comprises, at each imaginary-time step of the imaginary-time propagation: Transmitting, by the classical computer, data representing the wave function for the previous imaginary-time step to the quantum computer; Calculating, by the quantum computer, an inner product using the data representing the wave function for the previous imaginary-time step and the data representing the trial wave function; Transmitting, by the quantum computer, data representing the calculated inner product to the classical computer; Updating, by the classical computer, the wave function for the previous imaginary-time step using the data representing the calculated inner product to obtain a wave function for the current imaginary-time step.

13. The step of performing a measurement operation on the transformation of the copy of the trial wave function comprises: 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; Measuring the rotated trial wave function on a computational basis, the method according to claim 12. **Claim 14** The step of performing a measurement operation on the transformation of the 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 includes a qubit Clifford gate, and N p∈P represents the number of qubits in sub-part p of a partition of N qubits into P parts, and the step Comprises applying a tensor product of the randomly sampled unitary operator to the copy of the trial wave function to obtain a transformed trial wave function; Measuring the transformed trial wave function on a computational basis, the method according to claim 12. **Claim 15** The quantum Monte Carlo simulation includes a projective quantum Monte Carlo simulation or an auxiliary field quantum Monte Carlo simulation, the method according to claim 1. **Claim 16** The quantum computer includes a noisy intermediate-scale quantum device, the method according to claim 1. **Claim 17** The trial wave function includes a wave function from a generalized valence bond complete pairing wave function hypothesis, the method according to claim 1. **Claim 18** 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, the method according to claim 17. **Claim 19** One or more computers; One or more computer-readable recording media coupled to the one or more computers storing instructions that, when executed by the one or more computers, cause the one or more computers to perform the operations executed by the classical computer among the operations according to the method of any one of claims 1 to 11 and 15 to 18, a system. **Claim 20** One or more quantum computers; One or more computer-readable media coupled to the one or more quantum computers storing instructions that, when executed by the one or more quantum computers, cause the one or more quantum computers to perform the operations executed by the quantum computer among the operations according to the method of any one of claims 12 to 18, a system. **Claim 21** The quantum computer includes a NISQ device, the system according to claim 20.

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