Optimized numerical gradient and Hessian estimation for variational quantum algorithms

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