Phase shift rule with the optimal parameter selection

TL;DR

This paper introduces an optimized phase shift rule for quantum systems that accurately estimates derivatives regardless of eigenvalue spacing, reducing resource requirements in quantum simulations.

Contribution

It presents a novel, adaptable parameter shift rule that works for systems with closely spaced eigenvalues and minimizes gate resource usage.

Findings

Effective derivative estimation for arbitrary spectral configurations

Reduced number of phase shifts needed for accurate calculations

Applicable to complex quantum systems with dense spectra

Abstract

The phase shift rules enable the estimation of the derivative of a quantum state with respect to phase parameters, providing valuable insights into the behavior and dynamics of quantum systems. This capability is essential in quantum simulation tasks where understanding the behavior of complex quantum systems is of interest, such as simulating chemical reactions or condensed matter systems. However, parameter shift rules are typically designed for Hamiltonian systems with equidistant eigenvalues. For systems with closely spaced eigenvalues, effective rules have not been established. We provide insights about the optimal design of a parameter shift rule, tailored to various sorts of spectral information that may be available. The proposed method lets derivatives be calculated for any system, regardless of how close the eigenvalues are to each other. It also optimizes the number of phase shifts, which reduces the amount of gate resources needed.