Optimized numerical gradient and Hessian estimation for variational quantum algorithms

TL;DR

This paper develops optimized numerical estimators for gradient and Hessian calculations in variational quantum algorithms, reducing estimation errors and revealing exponential improvements with circuit size and sampling resources.

Contribution

It introduces operationally optimized finite-difference and scaled parameter-shift estimators, demonstrating their exponential error reduction and practical advantages over standard methods.

Findings

Estimation errors decrease exponentially with circuit qubits for optimized estimators.

A critical sampling-copy number exists where optimized difference estimators outperform standard parameter-shift.

Scaled parameter-shift estimators outperform unscaled ones in accuracy across various sampling conditions.

Abstract

Sampling noisy intermediate-scale quantum devices is a fundamental step that converts coherent quantum-circuit outputs to measurement data for running variational quantum algorithms that utilize gradient and Hessian methods in cost-function optimization tasks. This step, however, introduces estimation errors in the resulting gradient or Hessian computations. To minimize these errors, we discuss tunable numerical estimators, which are the finite-difference (including their generalized versions) and scaled parameter-shift estimators [introduced in Phys. Rev. A 103, 012405 (2021)], and propose operational circuit-averaged methods to optimize them. We show that these optimized numerical estimators offer estimation errors that drop exponentially with the number of circuit qubits for a given sampling-copy number, revealing a direct compatibility with the barren-plateau phenomenon. In particular, there exists a critical sampling-copy number below which an optimized difference estimator gives a smaller average estimation error in contrast to the standard (analytical) parameter-shift estimator, which exactly computes gradient and Hessian components. Moreover, this critical number grows exponentially with the circuit-qubit number. Finally, by forsaking analyticity, we demonstrate that the scaled parameter-shift estimators beat the standard unscaled ones in estimation accuracy under any situation, with comparable performances to those of the difference estimators within significant copy-number ranges, and are the best ones if larger copy numbers are affordable.