  Although tensor networks are powerful tools for simulating low-dimensional
quantum physics, tensor network algorithms are very computationally costly in
higher spatial dimensions. We introduce quantum gauge networks: a different
kind of tensor network ansatz for which the computation cost of simulations
does not explicitly increase for larger spatial dimensions. We take inspiration
from the gauge picture of quantum dynamics, which consists of a local
wavefunction for each patch of space, with neighboring patches related by
unitary connections. A quantum gauge network (QGN) has a similar structure,
except the Hilbert space dimensions of the local wavefunctions and connections
are truncated. We describe how a QGN can be obtained from a generic
wavefunction or matrix product state (MPS). All $2k$-point correlation
functions of any wavefunction for $M$ many operators can be encoded exactly by
a QGN with bond dimension $O(M^k)$. In comparison, for just $k=1$, an
exponentially larger bond dimension of $2^{M/6}$ is generically required for an
MPS of qubits. We provide a simple QGN algorithm for approximate simulations of
quantum dynamics in any spatial dimension. The approximate dynamics can achieve
exact energy conservation for time-independent Hamiltonians, and spatial
symmetries can also be maintained exactly. We benchmark the algorithm by
simulating the quantum quench of fermionic Hamiltonians in up to three spatial
dimensions.
