  Quantum computers have the potential to explore the vast Hilbert space of
entangled states that play an important role in the behavior of strongly
interacting matter. This opportunity motivates reconsidering the Hamiltonian
formulation of gauge theories, with a suitable truncation scheme to render the
Hilbert space finite-dimensional. Conventional formulations lead to a Hilbert
space largely spanned by unphysical states; given the current inability to
perform large scale quantum computations, we examine here how one might
restrict wave function evolution entirely or mostly to the physical subspace.
We consider such constructions for the simplest of these theories containing
dynamical gauge bosons -- U(1) lattice gauge theory without matter in $d=2,3$
spatial dimensions -- and find that electric-magnetic duality naturally plays
an important role. We conclude that this approach is likely to significantly
reduce computational overhead in $d=2$ by a reduction of variables and by
allowing one to regulate magnetic fluctuations instead of electric. The former
advantage does not exist in $d=3$, but the latter might be important for
asymptotically-free gauge theories.
