  Due to rapidly improving quantum computing hardware, Hamiltonian simulations
of relativistic lattice field theories have seen a resurgence of attention.
This computational tool requires turning the formally infinite-dimensional
Hilbert space of the full theory into a finite-dimensional one. For gauge
theories, a widely-used basis for the Hilbert space relies on the
representations induced by the underlying gauge group, with a truncation that
keeps only a set of the lowest dimensional representations. This works well at
large bare gauge coupling, but becomes less efficient at small coupling, which
is required for the continuum limit of the lattice theory. In this work, we
develop a new basis suitable for the simulation of an SU(2) lattice gauge
theory in the maximal tree gauge. In particular, we show how to perform a
Hamiltonian truncation so that the eigenvalues of both the magnetic and
electric gauge-fixed Hamiltonian are mostly preserved, which allows for this
basis to be used at all values of the coupling. Little prior knowledge is
assumed, so this may also be used as an introduction to the subject of
Hamiltonian formulations of lattice gauge theories.
