Extended parameter shift rules with minimal derivative variance for parameterized quantum circuits

TL;DR

This paper introduces an extended parameter shift rule (EPSR) for quantum circuit derivatives, optimizing shifts to reduce variance and handle complex operators, improving accuracy in quantum gradient computations.

Contribution

The paper proposes EPSR, a generalized parameter shift rule that minimizes derivative variance and applies to arbitrary Hermitian operators in quantum circuits.

Findings

EPSR provides an infinite set of shifts for optimal variance reduction.

EPSR extends PSRs to handle arbitrary Hermitian operators.

Numerical simulations confirm improved derivative accuracy with EPSR.

Abstract

Parameter shift rules (PSRs) are useful methods for computing arbitrary-order derivatives of the cost function in parameterized quantum circuits. The basic idea of PSRs is to evaluate the cost function at different parameter shifts, then use specific coefficients to combine them linearly to obtain the exact derivatives. In this work, we propose an extended parameter shift rule (EPSR) which generalizes a broad range of existing PSRs and has the following two advantages. First, EPSR offers an infinite number of possible parameter shifts, allowing the selection of the optimal parameter shifts to minimize the final derivative variance and thereby obtaining the more accurate derivative estimates with limited quantum resources. Second, EPSR extends the scope of the PSRs in the sense that EPSR can handle arbitrary Hermitian operator $H$ in gate $U(x) = \exp (iHx)$ in the parameterized quantum circuits, while existing PSRs are valid only for simple Hermitian generators $H$ such as simple Pauli words. Additionally, we show that the widely used ``general PSR'', introduced by Wierichs et al. (2022), is a special case of our EPSR, and we prove that it yields globally optimal shifts for minimizing the derivative variance under the weighted-shot scheme. Finally, through numerical simulations, we demonstrate the effectiveness of EPSR and show that the usage of the optimal parameter shifts indeed leads to more accurate derivative estimates.