Ground state preparation in $(2+1)$-dimensional pure $\mathbb{Z}_2$ lattice gauge theory via deterministic quantum imaginary time evolution

TL;DR

This paper demonstrates the application of deterministic quantum imaginary time evolution to efficiently find the ground state of a 2+1D $ ext{Z}_2$ lattice gauge theory, with high accuracy and reduced costs.

Contribution

It constructs gauge-invariant Pauli operators for the QITE algorithm, improving efficiency and accuracy in simulating lattice gauge theories.

Findings

Achieves less than 0.1% relative error for up to twelve-plaquette systems.

Reduces measurement and gate costs without adding extra errors.

Validates the approach through tensor network simulations and comparison with DMRG.

Abstract

In this paper, we apply the deterministic quantum imaginary time evolution (QITE) algorithm to obtain the ground state of a $2+1$-dimensional pure $\mathbb{Z}_2$ lattice gauge theory. We first construct the set of Pauli operators commuting with Gauss's law constraints, generalizing a previous result. This makes the deterministic QITE gauge-invariant and reduces both the measurement and gate costs significantly without adding extra algorithm errors in the QITE. Then, the classical numerical simulation of the deterministic QITE using tensor networks is performed, and the results are compared with the density matrix renormalization group (DMRG) to evaluate the accuracy of the algorithm. Specifically, we investigate the coupling and system size dependence, and find that the deterministic QITE can achieve a relative error of less than $0.1\%$ up to a twelve-plaquette system and coupling values in a regime that we study. Furthermore, the error dependence on the number of time steps is studied and discussed.