Approximating exponentials of commutators by optimized product formulas

TL;DR

This paper develops optimized product formulas to efficiently approximate the exponential of commutators, reducing computational cost and enhancing quantum simulation techniques.

Contribution

It introduces new optimized product formulas of orders 3 to 6 for approximating commutator exponentials, improving efficiency over previous methods.

Findings

Reduced number of exponentials needed for approximation

Enhanced efficiency in quantum control protocols

Applicable to various quantum simulation algorithms

Abstract

Trotter product formulas constitute a cornerstone quantum Hamiltonian simulation technique. However, the efficient implementation of Hamiltonian evolution of nested commutators remains an under explored area. In this work, we construct optimized product formulas of orders 3 to 6 approximating the exponential of a commutator of two arbitrary operators in terms of the exponentials of the operators involved. The new schemes require a reduced number of exponentials and thus provide more efficient approximations than other previously published alternatives. They can also be used as basic methods in recursive procedures to increase the order of approximation. We expect this research will improve the efficiency of quantum control protocols, as well as quantum algorithms such as the Zassenhaus-based product formula, Magnus operator-based time-dependent simulation, and product formula schemes with modified potentials.