  Combinatorial optimization on near-term quantum devices is a promising path
to demonstrating quantum advantage. However, the capabilities of these devices
are constrained by high noise levels and limited error mitigation. In this
paper, we propose an iterative Layer VQE (L-VQE) approach, inspired by the
Variational Quantum Eigensolver (VQE). We present a large-scale numerical
study, simulating circuits with up to 40 qubits and 352 parameters, that
demonstrates the potential of the proposed approach. We evaluate quantum
optimization heuristics on the problem of detecting multiple communities in
networks, for which we introduce a novel qubit-frugal formulation. We
numerically compare L-VQE with QAOA and demonstrate that QAOA achieves lower
approximation ratios while requiring significantly deeper circuits. We show
that L-VQE is more robust to sampling noise and has a higher chance of finding
the solution as compared with standard VQE approaches. Our simulation results
show that L-VQE performs well under realistic hardware noise.
