  In the near-future noisy intermediate-scale quantum (NISQ) era of quantum
computing technology, applications of quantum computing will be limited to
calculations of very modest scales in terms of the number of qubits used. The
need to represent numeric quantities using limited resources leads to
digitization errors which must be taken into account. As a first step towards
quantum simulations of realistic high-energy physics problems, we explore
classically the effects of digitizing elements of the $\mathrm{SU}(2)$ gauge
group to a finite set. We consider several methods for digitizing the group,
finding the best performance from an action-preserving projection onto a mesh.
Working in (3+1) dimensions, we find that using $\sim 7$ (qu)bits to represent
each $\mathrm{SU}(2)$ gauge link induces a digitization error on the order of
$10\%$ in short-distance observables and $2\%$ in long-distance observables.
Promisingly, our results indicate that each $\mathrm{SU}(2)$ gauge link can be
represented by $\mathcal{O}(10)$ (qu)bits, from which we estimate that a $16^3$
$\mathrm{SU}(2)$ lattice could be simulated with no more than
$\mathcal{O}(10^5)$ (qu)bits. Our results on digitization are also of interest
as a form of lossy compression that could be used in high-performance classical
computing to alleviate communications bottlenecks.
