Estimating the gradient and higher-order derivatives on quantum hardware

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