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 $mathrmSU(2)$ 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 $mathrmSU(2)$ 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 $mathrmSU(2)$ gauge link can be represented by $mathcalO(10)$ (qu)bits, from which we estimate that a $16textasciicircum3$ $mathrmSU(2)$ lattice could be simulated with no more than $mathcalO(10textasciicircum5)$ (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.