  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.
