Quantum Simulation of non-Abelian Lattice Gauge Theories: a variational approach to $\mathbb{D}_8$

TL;DR

This paper presents a resource-efficient variational quantum simulation approach for non-Abelian lattice gauge theories, specifically for the group $ ext{D}_8$, by removing matter and mapping to qudit systems for implementation on quantum hardware.

Contribution

It introduces a matter-removing procedure for non-Abelian lattice gauge theories and maps the theory onto qudit systems for variational quantum simulation.

Findings

Effective matter-removing procedure reduces quantum resource requirements.

Successful mapping of $ ext{D}_8$ gauge theory onto qudit systems.

Demonstrated feasibility for 1D and 2D lattice simulations.

Abstract

In this work, we address the problem of a resource-efficient formulation of non-Abelian LGTs by focusing on the difficulty of simulating fermionic degrees of freedom and the Hilbert space redundancy. First, we show a procedure that removes the matter and improves the efficiency of the hardware resources. We demonstrate it for the simplest non-Abelian group addressable with this procedure, $\mathbb{D}_8$, both in the cases of one (1D) and two (2D) spatial dimensions. Then, with the objective of running a variational quantum simulation on real quantum hardware, we map the $\mathbb{D}_8$ lattice gauge theory onto qudit systems with local interactions. We propose a variational scheme for the qudit system with a local Hamiltonian, which can be implemented on a universal qudit quantum device as the one developed in $\href{https://doi.org/10.1038/s41567-022-01658-0}{[Nat. Phys. 18, 1053 (2022)]}$. Our results show the effectiveness of the matter-removing procedure, solving the redundancy problem and reducing the amount of quantum resources. This can serve as a way of simulating lattice gauge theories in high spatial dimensions, with non-Abelian gauge groups, and including dynamical fermions.