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arXiv:2108.01218 · Physical Review A

Generalized quantum circuit differentiation rules

Oleksandr Kyriienko, Vincent E. Elfving

Published Nov 15, 2021• 52 citations
arXivHTMLPDF

Abstract

Variational quantum algorithms that are used for quantum machine learning rely on the ability to automatically differentiate parametrized quantum circuits with respect to underlying parameters. Here we propose the rules for differentiating quantum circuits (unitaries) with arbitrary generators. Unlike the standard parameter-shift rule valid for unitaries generated by operators with spectra limited to at most two unique eigenvalues (represented by involutory and idempotent operators), our approach also works for generators with a generic nondegenerate spectrum. Based on a spectral decomposition, we derive a simple recipe that allows explicit derivative evaluation. The derivative corresponds to the weighted sum of measured expectations for circuits with shifted parameters. The number of function evaluations is equal to the number of unique positive nonzero spectral gaps (eigenvalue differences) for the generator. We apply the approach to relevant examples of two-qubit gates, among others showing that the so-called fermionic simulation (fSim) gate can be differentiated using four measurements. Additionally, we present generalized differentiation rules for the case of Pauli-string generators, based on distinct shifts (here referred to as the triangulation approach), and analyze the variance for derivative measurements in different scenarios. Our work offers a toolbox for the efficient hardware-oriented differentiation needed for circuit optimization and operator-based derivative representation.

DOI
10.1103/physreva.104.052417

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Variational quantum algorithms that are used for quantum machine learning rely on the ability to automatically differentiate parametrized quantum circuits with respect to underlying parameters. Here we propose the rules for differentiating quantum circuits (unitaries) with arbitrary generators. Unlike the standard parameter-shift rule valid for unitaries generated by operators with spectra limited to at most two unique eigenvalues (represented by involutory and idempotent operators), our approach also works for generators with a generic nondegenerate spectrum. Based on a spectral decomposition, we derive a simple recipe that allows explicit derivative evaluation. The derivative corresponds to the weighted sum of measured expectations for circuits with shifted parameters. The number of function evaluations is equal to the number of unique positive nonzero spectral gaps (eigenvalue differences) for the generator. We apply the approach to relevant examples of two-qubit gates, among others showing that the so-called fermionic simulation (fSim) gate can be differentiated using four measurements. Additionally, we present generalized differentiation rules for the case of Pauli-string generators, based on distinct shifts (here referred to as the triangulation approach), and analyze the variance for derivative measurements in different scenarios. Our work offers a toolbox for the efficient hardware-oriented differentiation needed for circuit optimization and operator-based derivative representation.
Abstract Gradient-based optimization is a key ingredient of variational quantum algorithms, with applications ranging from quantum machine learning to quantum chemistry and simulation. The parameter-shift rule provides a hardware-friendly method for evaluating gradients of expectation values with respect to circuit parameters, but its applicability is limited to circuits whose gate generators have a particular spectral structure. In this work, we present a generalized framework that, with optimal minimum measurement overhead, extends parameter shift rules beyond this restrictive setting to encompass basically arbitrary gate generator, possibly made of complicated multi-qubit interactions with unknown spectrum and, in some settings, even infinite dimensional systems such as those describing photonic devices or qubit-oscillator systems. Our generalization enables the use of more expressive quantum circuits in variational quantum optimization and enlarges its scope by harnessing all the available hardware degrees of freedom.
Variational quantum algorithms are ubiquitous in applications of noisy intermediate-scale quantum computers. Due to the structure of conventional parametrized quantum gates, the evaluated functions typically are finite Fourier series of the input parameters. In this work, we use this fact to derive new, general parameter-shift rules for single-parameter gates, and provide closed-form expressions to apply them. These rules are then extended to multi-parameter quantum gates by combining them with the stochastic parameter-shift rule. We perform a systematic analysis of quantum resource requirements for each rule, and show that a reduction in resources is possible for higher-order derivatives. Using the example of the quantum approximate optimization algorithm, we show that the generalized parameter-shift rule can reduce the number of circuit evaluations significantly when computing derivatives with respect to parameters that feed into many gates. Our approach additionally reproduces reconstructions of the evaluated function up to a chosen order, leading to known generalizations of the Rotosolve optimizer and new extensions of the quantum analytic descent optimization algorithm.
Optimization of unitary transformations in variational quantum algorithms benefits highly from efficient evaluation of cost function gradients with respect to amplitudes of unitary generators. We propose several extensions of the parameter-shift rule to formulating these gradients as linear combinations of expectation values for generators with general eigenspectra (i.e., with more than two eigenvalues). Our approaches are exact and do not use any auxiliary qubits; instead they rely on a generator eigenspectrum analysis. Two main directions in the parameter-shift-rule extensions are (1) polynomial expansion of the exponential unitary operator based on a limited number of different eigenvalues in the generator and (2) decomposition of the generator as a linear combination of low-eigenvalue operators (e.g., operators with only two or three eigenvalues). These techniques have a range of scalings for the number of needed expectation values with the number of generator eigenvalues from quadratic (for polynomial expansion) to linear and even log2{log}_{2}log2​ (for generator decompositions). This allowed us to propose efficient differentiation schemes for commonly used two-qubit transformations (e.g., match gates, transmon gates, and fSim gates) and ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{2}$-conserving fermionic operators for the variational quantum eigensolver.
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.
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.