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arXiv:2207.01587 · Quantum

"Proper" Shift Rules for Derivatives of Perturbed-Parametric Quantum Evolutions

Dirk Oliver Theis

Published Jul 11, 2023• 2 citations
arXivHTMLPDF

Abstract

Banchi & Crooks (Quantum, 2021) have given methods to estimate derivatives of expectation values depending on a parameter that enters via what we call a "perturbed" quantum evolution x↦ei(xA+B)/ℏ . Their methods require modifications, beyond merely changing parameters, to the unitaries that appear. Moreover, in the case when the B -term is unavoidable, no exact method (unbiased estimator) for the derivative seems to be known: Banchi & Crooks's method gives an approximation.In this paper, for estimating the derivatives of parameterized expectation values of this type, we present a method that only requires shifting parameters, no other modifications of the quantum evolutions (a "proper" shift rule). Our method is exact (i.e., it gives analytic derivatives, unbiased estimators), and it has the same worst-case variance as Banchi-Crooks's.Moreover, we discuss the theory surrounding proper shift rules, based on Fourier analysis of perturbed-parametric quantum evolutions, resulting in a characterization of the proper shift rules in terms of their Fourier transforms, which in turn leads us to non-existence results of proper shift rules with exponential concentration of the shifts. We derive truncated methods that exhibit approximation errors, and compare to Banchi-Crooks's based on preliminary numerical simulations.

DOI
10.22331/q-2023-07-11-1052

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Banchi & Crooks (Quantum, 2021) have given methods to estimate derivatives of expectation values depending on a parameter that enters via what we call a "perturbed" quantum evolution x↦ei(xA+B)/ℏ . Their methods require modifications, beyond merely changing parameters, to the unitaries that appear. Moreover, in the case when the B -term is unavoidable, no exact method (unbiased estimator) for the derivative seems to be known: Banchi & Crooks's method gives an approximation.In this paper, for estimating the derivatives of parameterized expectation values of this type, we present a method that only requires shifting parameters, no other modifications of the quantum evolutions (a "proper" shift rule). Our method is exact (i.e., it gives analytic derivatives, unbiased estimators), and it has the same worst-case variance as Banchi-Crooks's.Moreover, we discuss the theory surrounding proper shift rules, based on Fourier analysis of perturbed-parametric quantum evolutions, resulting in a characterization of the proper shift rules in terms of their Fourier transforms, which in turn leads us to non-existence results of proper shift rules with exponential concentration of the shifts. We derive truncated methods that exhibit approximation errors, and compare to Banchi-Crooks's based on preliminary numerical simulations.
Parameter shift rules enable the estimation of the derivatives of expectation values with respect to the dynamical evolution of a state. Thus, they provide a valuable tool in variational optimization and insight into the dynamical behavior of quantum systems. Constructing parameter shift rules applicable to complex dynamical generators is useful towards improving the control of analog quantum simulation devices and extending the applicability of variational quantum simulating methods. However, parameter shift rules are typically designed only for dynamical generators with highly degenerate or equidistant eigenvalues, such as Pauli operators, commonplace in the construction of parametrized quantum circuits. For generators with irregularly spaced eigenvalues, effective rules have not been established. We provide insights about the optimal design of a parameter shift rule tailored to various sorts of spectral information that may be available. The proposed method targets the calculation of any linear combination of kthk\phantom{\rule{0.16em}{0ex}}\mathrm{th}kth derivatives with respect to the conjugate variable of a generator with a known spectrum, regardless of how close the eigenvalues are to each other. We discuss the optimal choice of parameter shifts, minimizing their number and the variance of the target estimator. Furthermore, we discuss the construction of parameter shift rules when only partial information about the generator spectrum is known.
Parameter shift rules (PSRs) are useful methods for computing arbitrary-order derivatives of the cost function in parameterized quantum circuits. The basic idea of PSRs is to evaluate the cost function at different parameter shifts, then use specific coefficients to combine them linearly to obtain the exact derivatives. In this work, we propose an extended parameter shift rule (EPSR) which generalizes a broad range of existing PSRs and has the following two advantages. First, EPSR offers an infinite number of possible parameter shifts, allowing the selection of the optimal parameter shifts to minimize the final derivative variance and thereby obtaining the more accurate derivative estimates with limited quantum resources. Second, EPSR extends the scope of the PSRs in the sense that EPSR can handle arbitrary Hermitian operator HHH in gate U(x)=exp⁡(iHx)U(x) = \exp (iHx)U(x)=exp(iHx) in the parameterized quantum circuits, while existing PSRs are valid only for simple Hermitian generators HHH such as simple Pauli words. Additionally, we show that the widely used ``general PSR'', introduced by Wierichs et al. (2022), is a special case of our EPSR, and we prove that it yields globally optimal shifts for minimizing the derivative variance under the weighted-shot scheme. Finally, through numerical simulations, we demonstrate the effectiveness of EPSR and show that the usage of the optimal parameter shifts indeed leads to more accurate derivative estimates.
Hybrid quantum-classical optimization algorithms represent one of the most promising application for near-term quantum computers. In these algorithms the goal is to optimize an observable quantity with respect to some classical parameters, using feedback from measurements performed on the quantum device. Here we study the problem of estimating the gradient of the function to be optimized directly from quantum measurements, generalizing and simplifying some approaches present in the literature, such as the so-called parameter-shift rule. We derive a mathematically exact formula that provides a stochastic algorithm for estimating the gradient of any multi-qubit parametric quantum evolution, without the introduction of ancillary qubits or the use of Hamiltonian simulation techniques. The gradient measurement is possible when the underlying device can realize all Pauli rotations in the expansion of the Hamiltonian whose coefficients depend on the parameter. Our algorithm continues to work, although with some approximations, even when all the available quantum gates are noisy, for instance due to the coupling between the quantum device and an unknown environment.
Variational (or, parameterized) quantum circuits are quantum circuits that contain real-number parameters, that need to be optimized/"trained" in order to achieve the desired quantum-computational effect. For that training, analytic derivatives (as opposed to numerical derivation) are useful. Parameter shift rules have received attention as a way to obtain analytic derivatives, via statistical estimators. In this paper, using Fourier Analysis, we characterize the set of all shift rules that realize a given fixed partial derivative of a multi-parameter VQC. Then, using Convex Optimization, we show how the search for the shift rule with smallest standard deviation leads to a primal-dual pair of convex optimization problems. We study these optimization problems theoretically, prove a strong duality theorem, and use it to establish optimal dual solutions for several families of VQCs. This also proves optimality for some known shift rules and answers the odd open question. As a byproduct, we demonstrate how optimal shift rules can be found efficiently computationally, and how the known optimal dual solutions help with that.
With the goal in mind of deriving a method to compute quantum corrections for the real-time evolution in quantum field theory, we analyze the problem from the perspective of the Wigner function. We argue that this provides the most natural way to justify and extend the classical approximation. A simple proposal is presented that can allow us to give systematic quantum corrections to the evolution of expectation values and/or an estimate of the errors committed when using the classical approximation. The method is applied to the case of a few degrees of freedom and compared with other methods and with the exact quantum results. An analysis of the dependence of the numerical effort involved as a function of the number of variables is given, which allows us to be optimistic about its applicability in a quantum field theoretical context.
Abstract Many near-term quantum computing algorithms are conceived as variational quantum algorithms, in which parameterized quantum circuits are optimized in a hybrid quantum–classical setup. Examples are variational quantum eigensolvers, quantum approximate optimization algorithms as well as various algorithms in the context of quantum-assisted machine learning. A common bottleneck of any such algorithm is constituted by the optimization of the variational parameters. A popular set of optimization methods work on the estimate of the gradient, obtained by means of circuit evaluations. We will refer to the way in which one can combine these circuit evaluations as gradient rules. This work provides a comprehensive picture of the family of gradient rules that vary parameters of quantum gates individually. The most prominent known members of this family are the parameter shift rule (PSR) and the finite differences method. To unite this family, we propose a generalized PSR that expresses all members of the aforementioned family as special cases, and discuss how all of these can be seen as providing access to a linear combination of exact first- and second-order derivatives. We further prove that a PSR with one non-shifted evaluation and only one shifted circuit evaluation does not exist, and introduce a novel perspective for approaching new gradient rules.