  We present a general framework for modifying quantum approximate optimization
algorithms (QAOA) to solve constrained network flow problems. By exploiting an
analogy between flow constraints and Gauss's law for electromagnetism, we
design lattice quantum electrodynamics (QED) inspired mixing Hamiltonians that
preserve flow constraints throughout the QAOA process. This results in an
exponential reduction in the size of the configuration space that needs to be
explored, which we show through numerical simulations, yields higher quality
approximate solutions compared to the original QAOA routine. We outline a
specific implementation for edge-disjoint path (EDP) problems related to
traffic congestion minimization, numerically analyze the effect of initial
state choice, and explore trade-offs between circuit complexity and qubit
resources via a particle-vortex duality mapping. Comparing the effect of
initial states reveals that starting with an ergodic (unbiased) superposition
of solutions yields better performance than beginning with the mixer
ground-state, suggesting a departure from the "short-cut to adiabaticity"
mechanism often used to motivate QAOA.
