  Recent theoretical and experimental studies have suggested that quantum Monte
Carlo (QMC) simulation can behave similarly to quantum annealing (QA). The
theoretical analysis was based on calculating transition rates between local
minima, in the large spin limit using WentzelKramers-Brillouin (WKB)
approximation, for highly symmetric systems of ferromagnetically coupled
qubits. The rate of transition was observed to scale the same in QMC and
incoherent quantum tunneling, implying that there might be no quantum advantage
of QA compared to QMC other than a prefactor. Quantum annealing is believed to
provide quantum advantage through large scale superposition and entanglement
and not just incoherent tunneling. Even for incoherent tunneling, the scaling
similarity with QMC observed above does not hold in general. Here, we compare
incoherent tunneling and QMC escape using perturbation theory, which has much
wider validity than WKB approximation. We show that the two do not scale the
same way when there are multiple homotopy-inequivalent paths for tunneling. We
demonstrate through examples that frustration can generate an exponential
number of tunneling paths, which under certain conditions can lead to an
exponential advantage for incoherent tunneling over classical QMC escape. We
provide analytical and numerical evidence for such an advantage and show that
it holds beyond perturbation theory.
