Digitization and subduction of $SU(N)$ gauge theories

TL;DR

This paper explores how digitizing $SU(N)$ gauge theories using discrete subgroups affects simulation accuracy, providing methods to identify and mitigate approximation errors through classical calculations and static potential analysis.

Contribution

It introduces a systematic approach to analyze the subduction of $SU(2)$ and $SU(3)$ to discrete subgroups, aiding in improving lattice gauge theory simulations on quantum computers.

Findings

Subduction provides valuable information for gauge field digitization.

Percent-level agreement with Casimir scaling for certain irreducible representations.

Some irreducible representations show significant deviations from expected results.

Abstract

The simulation of lattice gauge theories on quantum computers necessitates digitizing gauge fields. One approach involves substituting the continuous gauge group with a discrete subgroup, but the implications of this approximation still need to be clarified. To gain insights, we investigate the subduction of $ SU(2) $ and $ SU(3)$ to discrete crystal-like subgroups. Using classical lattice calculations, we show that subduction offers valuable information based on subduced direct sums, helping us identify additional terms to incorporate into the lattice action that can mitigate the effects of digitization. Furthermore, we compute the static potentials of all irreducible representations of $ \Sigma(360 \times 3) $ at a fixed lattice spacing. Our results reveal a percent-level agreement with the Casimir scaling of ( SU(3) ) for irreducible representations that subduce to a single $ \Sigma(360 \times 3) $ irreducible representation. This provides a diagnostic measure of approximation quality, as some irreducible representations closely match the expected results while others exhibit significant deviations.