Estimating the gradient and higher-order derivatives on quantum hardware

TL;DR

This paper develops analytical methods for efficiently computing derivatives of variational quantum circuits, including higher-order derivatives like the Hessian, enabling advanced optimization techniques on quantum hardware.

Contribution

It introduces parameter-shift rules for arbitrary-order derivatives and demonstrates their application to second-order optimization algorithms on quantum computers.

Findings

Parameter-shift rules for Hessian and metric tensor derived

Efficient implementation of second-order optimization algorithms

Numerical and hardware experiments validate methods

Abstract

For a large class of variational quantum circuits, we show how arbitrary-order derivatives can be analytically evaluated in terms of simple parameter-shift rules, i.e., by running the same circuit with different shifts of the parameters. As particular cases, we obtain parameter-shift rules for the Hessian of an expectation value and for the metric tensor of a variational state, both of which can be efficiently used to analytically implement second-order optimization algorithms on a quantum computer. We also consider the impact of statistical noise by studying the mean squared error of different derivative estimators. In the second part of this work, some of the theoretical techniques for evaluating quantum derivatives are applied to their typical use case: the implementation of quantum optimizers. We find that the performance of different estimators and optimizers is intertwined with the values of different hyperparameters, such as a step size or a number of shots. Our findings are supported by several numerical and hardware experiments, including an experimental estimation of the Hessian of a simple variational circuit and an implementation of the Newton optimizer.