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EP-4174732-B1 - ESTIMATING AN EXTREMUM OF THE EXPECTATION VALUE OF A RANDOM FIELD

EP4174732B1EP 4174732 B1EP4174732 B1EP 4174732B1EP-4174732-B1

Inventors

  • BOSSE, Jan Lukas

Dates

Publication Date
20260506
Application Date
20221028

Claims (14)

  1. A method of estimating an extremum of an expectation value of a random field, the method executed on a computer (200) and utilising a quantum computer (150), wherein the quantum computer comprises a quantum circuit parameterised by a set of circuit parameters and configured to estimate an expectation value of a random variable of interest for the set of circuit parameters, wherein the method comprises: storing a set of circuit parameters for the quantum circuit; storing a set of model parameters for a surrogate model, wherein the surrogate model is configured, based on the stored set of model parameters, to approximate the expectation value of the random variable of interest produced by the quantum computer for circuit parameters in the vicinity of the stored set of circuit parameters; storing uncertainties associated with each of the stored set of model parameters; performing an iterative optimisation to update the stored set of circuit parameters until a criterion is met indicating that the stored set of circuit parameters are approximate to circuit parameters representing an extremum of the expectation value of the random field, wherein the iterative optimisation comprises a plurality of iterations to update the stored set of circuit parameters, stored set of model parameters and stored uncertainties associated with the model parameters, wherein each iteration comprises: determining a plurality of sampling sets of circuit parameters within a sampling radius of the stored set of circuit parameters; for each of the plurality of sampling sets of circuit parameters, executing the quantum computer a plurality of times to determine estimates of the expectation value and the uncertainty of the estimate calculated by the quantum computer for each of the plurality of sampling sets of circuit parameters, wherein the estimate of the uncertainty of the expectation value is determined based on a variance of noisy results returned by the quantum computer when executed a plurality of times; determining an updated set of model parameters for the surrogate model in dependence on the estimates and uncertainties of the expectation values calculated for each of the sampling sets of circuit parameters, the stored model parameters and the stored uncertainties associated with the stored model parameters; determining interim uncertainties associated with the updated model parameters in dependence on the uncertainty of the expectation values calculated for each of the sampling sets of circuit parameters and the stored uncertainties associated with the stored model parameters; determining a gradient of the surrogate model at the stored circuit parameters when parameterised by the updated set of model parameters; determining an updated set of circuit parameters in dependence on the stored set of circuit parameters and the determined gradient of the surrogate model, wherein the updated set of circuit parameters represents a descent or ascent along the determined gradient of the surrogate model, relative to the stored set of circuit parameters; determining updated uncertainties associated with the updated model parameters for the updated set of circuit parameters, wherein the updated uncertainties are determined in dependence on the determined interim uncertainties and a step size of the descent or ascent along the gradient of the surrogate model used to determine the updated set of circuit parameters; updating the stored circuit parameters to correspond to the determined updated set of circuit parameters; updating the stored set of model parameters to correspond to the determined updated set of model parameters; and updating the stored uncertainties associated with the model parameters to correspond to the determined updated uncertainties.
  2. The method of claim 1, wherein the quantum computer (150) is configured to estimate the energy of a state produced by the quantum circuit with respect to a given quantum Hamiltonian for a set of circuit parameters, and wherein the expectation value of the random variable estimated by the quantum computer (150) comprises the energy of a state produced by the quantum circuit for a set of circuit parameters.
  3. The method of claim 1 or 2, wherein the quantum computer (150) is configured to simulate a physical system, which may comprise a quantum mechanical system.
  4. The method of claim 3, wherein the method comprises a method of estimating a ground state energy of the physical system.
  5. The method of any preceding claim, wherein the surrogate model comprises a quadratic model.
  6. The method of any preceding claim, wherein the stored model parameters are each represented by a multivariate gaussian distribution.
  7. The method of claim 6, wherein the determining an updated set of model parameters for the surrogate model and determining interim uncertainties associated with the updated model parameters comprises performing a Bayesian update of gaussian distributions representing the stored model parameters: wherein optionally the stored set of model parameters and associated uncertainties represent a prior and the mean and variance of the expectation values calculated for each of the sampling sets of circuit parameters represent new data for performing the Bayesian update.
  8. The method of any preceding claim, wherein the step size of the ascent or descent along the gradient of the surrogate model used to determine the updated set of circuit parameters is proportional to the magnitude of the determined gradient.
  9. The method of any preceding claim, wherein determining updated uncertainties associated with the updated model parameters for the updated set of circuit parameters comprises adding additional uncertainty to the determined interim uncertainties in dependence on the step size of the descent or ascent along the gradient of the surrogate model used to determine the updated set of circuit parameters, wherein optionally the additional uncertainty is proportional to the squared step size of the descent or ascent along the gradient of the surrogate model used to determine the updated set of circuit parameters.
  10. The method of any preceding claim, wherein performing an iterative optimisation to update the stored set of circuit parameters until a criterion is met indicating that the stored set of circuit parameters are approximate to circuit parameters representing an extremum of the expectation value of the random field of interest comprises performing the iterative optimisation process until the step size of the descent or ascent along the gradient of the surrogate model used to determine the updated set of circuit parameters is less than a threshold step size.
  11. The method of any preceding claim, wherein performing an iterative optimisation to update the stored set of circuit parameters until a criterion is met indicating that the stored set of circuit parameters are approximate to circuit parameters representing an extremum of the expectation value of the random field of interest comprises performing the iterative optimisation process until the number of iterations performed reaches a threshold number of iterations.
  12. A computer readable medium having instructions stored thereon which, when read by a processor of a computing apparatus, cause the computing apparatus and a quantum computer to perform a method according to any preceding claim.
  13. A computing apparatus (200) comprising: one or more memories (220); one or more classical processors (210) configured to execute instructions stored in the one or more memory units to perform a method according to any of claims 1-11; and a quantum computer (150) configured to estimate an expectation value of the random variable of interest for a set of circuit parameters.
  14. A computing apparatus (200) according to claim 13, further comprising interaction means for interacting with the quantum computer.

Description

Technical Field The present disclosure relates to estimating an extremum of an expectation value of a random field. In particular, the present disclosure relates to optimising input parameters provided to a computing system, wherein the computing system is configured to estimate an expectation value, and the uncertainty of that estimate, of a random variable of interest for a set of input parameters. The present disclosure may find particular application in quantum computing. Background Advances in quantum computing have ushered in an era of noisy, intermediate-scale quantum (NISQ) hardware, which can no longer be simulated effectively classically, even on the world's largest supercomputers. One of the most important application areas of near-term quantum computers is predicted to be simulating quantum mechanical systems. Quantum computers could approximate values of physical quantities which are hard to obtain classically. However, this NISQ hardware is still extremely limited in terms of the number of qubits that can be controlled and the decoherence time - the time after which the fidelity of the qubits has degraded to the extent that the results of a performed algorithm are effectively meaningless. Currently quantum computing architectures have on the order of ≈ 50 qubits, and are only capable of implementing quantum circuits up to a depth of order ≈ 50 before decoherence renders the results meaningless. Critical factors that determine the feasibility of performing an algorithm include its qubit requirements (e.g. the number of qubits required and the ability to control interactions between them), its gate cost and its circuit depth (a measure of algorithm runtime). NISQ quantum computers are affected by noise and errors which can lead to highly inaccurate results being produced. Standard quantum fault-tolerance techniques introduce significant overheads in terms of both the number of qubits required and the associated gate cost, and so such techniques are unsuitable for the NISQ regime. To at least partially alleviate the burden of such challenges, several classical-quantum hybrid algorithms have been proposed. In a classical-quantum hybrid algorithm, both classical and quantum resources are used to perform a computational task. One non-trivial example is the Variational Quantum Eigensolver (VQE) algorithm, which uses quantum and classical resources to find variational solutions to eigenvalue and optimization problems that are not accessible to traditional classical computers. More specifically, the VQE algorithm is used to find eigenvalues of a (usually large) matrix H which may represent the Hamiltonian of a physical system. A quantum subroutine is run inside of a classical optimization loop. The quantum subroutine comprises preparing an ansatz quantum state |ψ(ω)) parametrised by a parameter ω and measuring the expectation value 〈ψ(ω)|H|ψ(ω)〉. The variational principle ensures that this expectation value is always greater than the smallest eigenvalue of the matrix H. In the classical optimization loop, the parameter ω can be iteratively adjusted until a convergence condition is met. The present invention has been devised in the foregoing context. "Verifying the output of quantum optimizers with ground-state energy lower bounds" Flavio Baccari et al, arxiv.org, Cornell University Library, 3 November 2020, DOI: 10.1103/PHYSREVRESEARCH.20.043163 relates to a consideration of the use of relaxations to a ground-state problem as a benchmark for the output of quantum optimisers. Summary The present invention is defined in the independent claims. Further embodiments of the invention are set forth in the dependent claims. One of the most prominent quantum algorithms in the Noisy Intermediate-Scale Quantum (NISQ) era [1] is the variational quantum eigensolver [2] (VQE). The VQE approach attempts to produce the ground (lowest-energy) state of a quantum or classical system by optimising over some family of quantum circuits. A quantum computer is used as a subroutine within an overall classical optimisation loop. The role of the quantum computer is to calculate the energy of the state produced by a parametrised quantum circuit. The information obtained from the energy measurements is used by the classical optimiser to update the parameters of the quantum circuit, and ultimately to find the lowest-energy state. This optimisation process is challenging as measurements are noisy, due both to statistical noise and to errors in the quantum hardware. Standard optimisation algorithms that assume exact knowledge of the objective function, such as Broyden-Fletcher-Goldfard-Shanno (BFGS) algorithm, struggle with this nonstandard setting. However, in the VQE setting the energy is estimated as the sample mean of many energy measurements, giving information about its uncertainty. This is unlike the standard setting for stochastic optimisation, where this information is not typically known. Here we introduce a new algorithm f