Generalized quantum circuit differentiation rules

TL;DR

This paper introduces a generalized method for differentiating quantum circuits with arbitrary generators, extending beyond the standard parameter shift rule, enabling more efficient quantum circuit optimization.

Contribution

The authors develop a spectral decomposition-based differentiation rule applicable to generators with non-degenerate spectra, broadening the scope of quantum circuit differentiation methods.

Findings

The method applies to generators with arbitrary spectra, not just involutory or idempotent.

For two-qubit gates like fSim, the derivative can be computed with four measurements.

Variance analysis of derivative measurements provides insights into measurement efficiency.

Abstract

Variational quantum algorithms that are used for quantum machine learning rely on the ability to automatically differentiate parametrized quantum circuits with respect to underlying parameters. Here, we propose the rules for differentiating quantum circuits (unitaries) with arbitrary generators. Unlike the standard parameter shift rule valid for unitaries generated by operators with spectra limited to at most two unique eigenvalues (represented by involutory and idempotent operators), our approach also works for generators with a generic non-degenerate spectrum. Based on a spectral decomposition, we derive a simple recipe that allows explicit derivative evaluation. The derivative corresponds to the weighted sum of measured expectations for circuits with shifted parameters. The number of function evaluations is equal to the number of unique positive non-zero spectral gaps (eigenvalue differences) for the generator. We apply the approach to relevant examples of two-qubit gates, among others showing that the fSim gate can be differentiated using four measurements. Additionally, we present generalized differentiation rules for the case of Pauli string generators, based on distinct shifts (here named as the triangulation approach), and analyse the variance for derivative measurements in different scenarios. Our work offers a toolbox for the efficient hardware-oriented differentiation needed for circuit optimization and operator-based derivative representation.