Efficient Product Formulas for Commutators and Applications to Quantum Simulation

TL;DR

This paper develops efficient product formulas for exponentials of commutators, improving quantum simulation methods and fidelity in state preparation, with applications to fermion chains and quantum Hall phases.

Contribution

It introduces a third-order product formula with fewer exponentials and recursive higher-order formulas, enhancing efficiency in quantum simulation.

Findings

Reduced gate count for desired accuracy

Inclusion of linear terms without extra cost

Enhanced fidelity in quantum state preparation

Abstract

We construct product formulas for exponentials of commutators and explore their applications. First, we directly construct a third-order product formula with six exponentials by solving polynomial equations obtained using the operator differential method. We then derive higher-order product formulas recursively from the third-order formula. We improve over previous recursive constructions, reducing the number of gates required to achieve the same accuracy. In addition, we demonstrate that the constituent linear terms in the commutator can be included at no extra cost. As an application, we show how to use the product formulas in a digital protocol for counterdiabatic driving, which increases the fidelity for quantum state preparation. We also discuss applications to quantum simulation of one-dimensional fermion chains with nearest- and next-nearest-neighbor hopping terms, and two-dimensional fractional quantum Hall phases.