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.
Highlights
The possibility of a quantum speedup for finding the solution of classical optimization problems is tantalizing, as a quantum advantage for this class of problems would provide a wealth of new applications for quantum computing
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 quantum annealing (QA)
We find numerically that adiabatic QA (AQA) needs a time of Oðn0.5Þ to find the ground state, where n is the number of spins or qubits
Summary
The possibility of a quantum speedup for finding the solution of classical optimization problems is tantalizing, as a quantum advantage for this class of problems would provide a wealth of new applications for quantum computing. We analyze the spin-coherent potential for several examples from a well-known class of problems known as perturbed Hamming weight oracle (PHWO) problems These are problems for which instances can be generated where QA either has an advantage over classical random search algorithms with local updates, such as SA [12,16], or has no advantage [16,17]. Having obtained a clear picture of tunneling, we focus on a particular PHWO problem that has a plateau in the final cost function (,“the fixed plateau”). We have essentially an arbitrary polynomial tunneling speedup of QA over SA on a cost function that is neither tall nor thin We remark that this result about SA’s performance is a rigorous proof of a result due to Reichardt [16] that classical local search algorithms will fail on a certain class of PHWO problems and is of independent interest. Additional background information and technical details can be found in the Appendixes
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