  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 or error rates. 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
Quantum Approximate Optimization Algorithm (QAOA) and demonstrate that QAOA
achieves lower approximation ratios while requiring significantly deeper
circuits. We show that L-VQE is more robust to finite sampling errors 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.
