  We present a digitization scheme for the lattice $\mathrm{SU}(2)$ gauge
theory Hamiltonian in the $\mathit{magnetic}$ $\mathit{basis}$, where the gauge
links are unitary and diagonal. The digitization is obtained from a particular
partitioning of the $\mathrm{SU}(2)$ group manifold, with the canonical momenta
constructed by an approximation of the Lie derivatives on this partitioning.
This construction, analogous to a discrete Fourier transform, preserves the
spectrum of the kinetic part of the Hamiltonian and the canonical commutation
relations exactly on a subspace of the truncated Hilbert space, while the
residual subspace can be projected above the cutoff of the theory.
