Physics

Tensor (intrinsic definition)

Metric (unit)

Quantum phase estimation algorithm

Pure mathematics

Mathematics

Quantum error correction

Algorithm

Statistics

Mathematical optimization

Epistemology

Quantum algorithm

Quantum circuit

Philosophy

Estimator

Economics

Hessian matrix

Divergence (linguistics)

Operations management

Computer science

Linguistics

Simple (philosophy)

Quantum computer

Quantum mechanics

Applied mathematics

Quantum

Hyperparameter

Physical review. A/Physical review, A

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.

Estimating the gradient and higher-order derivatives on quantum hardware