Optimized numerical gradient and Hessian estimation for variational quantum algorithms

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.

24 pages, 7 figures (updated Fig. 4, new Fig. 6, new Secs. IV C, V C, VII and Appendix C5 since last version)

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Cited by in corpus (2)

• Pitfalls when tackling the exponential concentration of parameterized quantum models
• Robustness of optimized numerical estimation schemes for noisy variational quantum algorithms