Tunneling and speedup in quantum optimization for permutation-symmetric problems

TL;DR

This paper investigates quantum tunneling in quantum annealing for permutation-symmetric problems, revealing that diabatic transitions can outperform adiabatic tunneling and classical algorithms, challenging common assumptions about quantum speedup mechanisms.

Contribution

It demonstrates that diabatic transitions, not tunneling, can lead to exponential speedups in quantum optimization, and compares their effectiveness to classical methods.

Findings

Diabatic transitions can outperform adiabatic tunneling in quantum annealing.

Classical spin vector dynamics can be as efficient as diabatic quantum annealing.

Speedups are observed even in problems with convex cost functions.

Abstract

Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), especially for problems featuring a cost function with tall and thin barriers. We present and analyze several counterexamples from the class of perturbed Hamming-weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semi-classical potential arising from the spin-coherent path integral formalism. We then provide an example where the shape of the barrier in the final cost function is short and wide, which might suggest no quantum advantage for QA, yet where tunneling renders QA superior to simulated annealing in the adiabatic regime. However, the adiabatic dynamics turn out not be optimal. Instead, an evolution involving a sequence of diabatic transitions through many avoided level-crossings, involving no tunneling, is optimal and outperforms adiabatic QA. We show that this phenomenon of speedup by diabatic transitions is not unique to this example, and we provide an example where it provides an exponential speedup over adiabatic QA. In yet another twist, we show that a classical algorithm, spin vector dynamics, is at least as efficient as diabatic QA. Finally, in a different example with a convex cost function, the diabatic transitions result in a speedup relative to both adiabatic QA with tunneling and classical spin vector dynamics.