  Gauge symmetries play a key role in physics appearing in areas such as
quantum field theories of the fundamental particles and emergent degrees of
freedom in quantum materials. Motivated by the desire to efficiently simulate
many-body quantum systems with exact local gauge invariance, gauge equivariant
neural-network quantum states are introduced, which exactly satisfy the local
Hilbert space constraints necessary for the description of quantum lattice
gauge theory with Zd gauge group on different geometries. Focusing on the
special case of Z2 gauge group on a periodically identified square lattice, the
equivariant architecture is analytically shown to contain the loop-gas solution
as a special case. Gauge equivariant neural-network quantum states are used in
combination with variational quantum Monte Carlo to obtain compact descriptions
of the ground state wavefunction for the Z2 theory away from the exactly
solvable limit, and to demonstrate the confining/deconfining phase transition
of the Wilson loop order parameter.
