Trotter errors in digital adiabatic quantum simulation of quantum $\mathbb{Z}_2$ lattice gauge theory

TL;DR

This paper investigates Trotter errors in digital quantum simulations of 2+1D $ ext{Z}_2$ lattice gauge theory, showing that symmetric decompositions yield significantly smaller errors and providing insights for experimental implementations.

Contribution

It provides a detailed analysis of Trotter errors in digital quantum simulations of $ ext{Z}_2$ lattice gauge theory, highlighting the advantages of symmetric decomposition.

Findings

Symmetric Trotter decomposition results in smaller errors than asymmetric.

Trotter errors are closely related to energy errors but are generally smaller than estimated.

Error behavior depends on the coupling parameter and number of Trotter steps.

Abstract

Trotter decomposition is the basis of the digital quantum simulation. Asymmetric and symmetric decompositions are used in our GPU demonstration of the digital adiabatic quantum simulations of $2+1$ dimensional quantum $\mathbb{Z}_2$ lattice gauge theory. The actual errors in Trotter decompositions are investigated as functions of the coupling parameter and the number of Trotter substeps in each step of the variation of coupling parameter. The relative error of energy is shown to be closely related to the Trotter error usually defined defined in terms of the evolution operators. They are much smaller than the order-of-magnitude estimation. The error in the symmetric decomposition is much smaller than that in the asymmetric decomposition. The features of the Trotter errors obtained here are useful in the experimental implementation of digital quantum simulation and its numerical demonstration.