  Quantum technologies offer the prospect to efficiently simulate sign-problem
afflicted regimes in lattice field theory, such as the presence of topological
terms, chemical potentials, and out-of-equilibrium dynamics. In this work, we
derive the (3+1)D topological $\theta$-term for Abelian and non-Abelian lattice
gauge theories in the Hamiltonian formulation, paving the way towards
Hamiltonian-based simulations of such terms on quantum and classical computers.
We further study numerically the zero-temperature phase structure of a (3+1)D
U(1) lattice gauge theory with the $\theta$-term via exact diagonalization for
a single periodic cube. In the strong coupling regime, our results suggest the
occurrence of a phase transition at constant values of $\theta$, as indicated
by an avoided level-crossing and abrupt changes in the plaquette expectation
value, the electric energy density, and the topological charge density. These
results could in principle be cross-checked by the recently developed (3+1)D
tensor network methods and quantum simulations, once sufficient resources
become available.
