Zero-temperature quantum annealing bottlenecks in the spin-glass phase.

Sergey Knysh

doi:10.1038/ncomms12370

Abstract

A promising approach to solving hard binary optimization problems is quantum adiabatic annealing in a transverse magnetic field. An instantaneous ground state—initially a symmetric superposition of all possible assignments of N qubits—is closely tracked as it becomes more and more localized near the global minimum of the classical energy. Regions where the energy gap to excited states is small (for instance at the phase transition) are the algorithm's bottlenecks. Here I show how for large problems the complexity becomes dominated by O(log N) bottlenecks inside the spin-glass phase, where the gap scales as a stretched exponential. For smaller N, only the gap at the critical point is relevant, where it scales polynomially, as long as the phase transition is second order. This phenomenon is demonstrated rigorously for the two-pattern Gaussian Hopfield model. Qualitative comparison with the Sherrington-Kirkpatrick model leads to similar conclusions.

Highlights

A case in point is the analysis of ref. 26, which develops perturbation theory in G
In the case of the former—and further, if the sensitivity of energy levels to changes in the transverse field is so large that the levels ‘collide’ before either valley disappears—
‘Zeeman splitting’ for G40 scales as N, which, for large problems, may be sufficient to overcome the O(1) classical gap and cause avoided crossings of levels associated with different classical energies. This trend disappears if only levels with the smallest energies are considered; these are relevant for avoided crossings with the ground state

Summary

Introduction

A case in point is the analysis of ref. 26, which develops perturbation theory in G. A case in point is the analysis of ref. The classical limit (G 1⁄4 0) is used as a starting point; how that analysis might be extended to G40 has been discussed[29]. A type of CSP has been considered: classical energy levels are discrete non-negative integers (number of violated constraints) but hpavffiffieffiffi exponential degeneracy. ‘Zeeman splitting’ for G40 scales as N, which, for large problems, may be sufficient to overcome the O(1) classical gap and cause avoided crossings of levels associated with different classical energies. This trend disappears if only levels with the smallest energies

Methods

Results

Conclusion