Quantum walks and non-Abelian discrete gauge theory

TL;DR

This paper introduces a new family of discrete-time quantum walks with exact $U(N)$ gauge invariance, connecting them to Dirac fermion dynamics in 2D and exploring their potential for simulating Yang-Mills theories.

Contribution

It presents a novel class of quantum walks with gauge invariance, their continuous limit, and numerical analysis, advancing quantum simulation of gauge theories.

Findings

Quantum walks exhibit exact $U(N)$ gauge invariance.

Continuous limit matches Dirac fermion coupled to $U(N)$ gauge fields.

Numerical simulations show convergence to continuous dynamics and compare with classical fields.

Abstract

A new family of discrete-time quantum walks (DTQWs) on the line with an exact discrete $U(N)$ gauge invariance is introduced. It is shown that the continuous limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac fermion coupled to usual $U(N)$ gauge fields in $2D$ spacetime. A discrete generalization of the usual $U(N)$ curvature is also constructed. An alternate interpretation of these results in terms of superimposed $U(1)$ Maxwell fields and $SU(N)$ gauge fields is discussed in the Appendix. Numerical simulations are also presented, which explore the convergence of the DTQWs towards their continuous limit and which also compare the DTQWs with classical (i.e. non-quantum) motions in classical $SU(2)$ fields. The results presented in this article constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs.