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arXiv:2106.01388 · Quantum Science and Technology

Single-component gradient rules for variational quantum algorithms

Thomas Hubregtsen, Frederik Wilde, Shozab Qasim, Jens Eisert

Published Apr 19, 2022• 11 citations
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Abstract

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.

DOI
10.1088/2058-9565/ac6824

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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.
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.
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.
An important application for near-term quantum computing lies in optimization tasks, with applications ranging from quantum chemistry and drug discovery to machine learning. In many settings, most prominently in so-called parametrized or variational algorithms, the objective function is a result of hybrid quantum-classical processing. To optimize the objective, it is useful to have access to exact gradients of quantum circuits with respect to gate parameters. This paper shows how gradients of expectation values of quantum measurements can be estimated using the same, or almost the same, architecture that executes the original circuit. It generalizes previous results for qubit-based platforms, and proposes recipes for the computation of gradients of continuous-variable circuits. Interestingly, in many important instances it is sufficient to run the original quantum circuit twice while shifting a single gate parameter to obtain the corresponding component of the gradient. More general cases can be solved by conditioning a single gate on an ancilla.
Many optimization methods for training variational quantum algorithms are based on estimating gradients of the cost function. Due to the statistical nature of quantum measurements, this estimation requires many circuit evaluations, which is a crucial bottleneck of the whole approach. We propose a new gradient estimation method to mitigate this measurement challenge and reduce the required measurement rounds. Within a Bayesian framework and based on the generalized parameter shift rule, we use prior information about the circuit to find an estimation strategy that minimizes expected statistical and systematic errors simultaneously. We demonstrate that this approach can significantly outperform traditional gradient estimation methods, reducing the required measurement rounds by up to an order of magnitude for a common QAOA setup. Our analysis also shows that an estimation via finite differences can outperform the parameter shift rule in terms of gradient accuracy for small and moderate measurement budgets.
Quantum computing is an active research interest in the new computational area with various applications. In the quantum machine learning field, the parameterized quantum circuit is a core learnable model; this model is updated iteratively by the general Parameter-Shift Rule (PSR) technique. In many qubits and parameter scenarios, using PSR consumes significant computational resources, particularly in the quantum simulator. Therefore, this research proposes a method to perform the PSR more efficiently for matrix multiplication and state-based real-world quantum simulators. The results show that our method is faster than baseline matrix multiplication-based 12 times and baseline state-based 39 times on average experiment.
Abstract In this study, we study the Variational Quantum Eigensolver (VQE) application for the Ising model as a test bed model, in which we pivotally delved into several optimization methods, both classical and quantum, and analyzed the quantum advantage that each of these methods offered, and then we proposed a new combinatorial optimization scheme, deemed as QN-SPSA+PSR which combines calculating approximately Fubini-study metric (QN-SPSA) and the exact evaluation of gradient by Parameter-Shift Rule (PSR). The QN-SPSA+PSR method integrates the QN-SPSA computational efficiency with the precise gradient computation of the PSR, improving both stability and convergence speed while maintaining low computational consumption. Our results provide a new potential quantum supremacy in the VQAs's optimization subroutine, even in Quantum Machine Learning's optimization section, and enhance viable paths toward efficient quantum simulations on Noisy Intermediate-Scale Quantum Computing (NISQ) devices. Additionally, we also conducted a detailed study of quantum circuit ansatz structures in order to find the one that would work best with the Ising model and NISQ, in which we utilized the properties of the investigated model.