Reverse annealing for the fully connected $p$-spin model

TL;DR

This paper develops a mean-field theory for reverse annealing applied to the fully-connected p-spin model, showing how it can effectively avoid first-order phase transitions that hinder conventional quantum annealing.

Contribution

It provides the first analytical study of reverse annealing in the p-spin model, revealing conditions under which it outperforms traditional quantum annealing.

Findings

Reverse annealing can bypass first-order phase transitions in the p-spin model.

Effectiveness depends on the initial state's proximity to the solution.

Smaller order parameter jumps improve tunneling and solution success.

Abstract

Reverse annealing is a variant of quantum annealing that starts from a given classical configuration of spins (qubits). In contrast to the conventional formulation, where one starts from a uniform superposition of all possible states (classical configurations), quantum fluctuations are first increased and only then decreased. One then reads out the state as a proposed solution to the given combinatorial optimization problem. We formulate a mean-field theory of reverse annealing using the fully-connected ferromagnetic $p$-spin model, with and without random longitudinal fields, and analyze it in order to understand how and when reverse annealing is effective at solving this problem. We find that the difficulty arising from the existence of a first-order quantum phase transition, which leads to an exponentially long computation time in conventional quantum annealing, is circumvented in the context of this particular problem by reverse annealing if the proximity of the initial state to the (known) solution exceeds a threshold. Even when a first-order transition is unavoidable, the difficulty is mitigated due to a smaller jump in the order parameter at a first-order transition, which implies a larger rate of quantum tunneling. This is the first analytical study of reverse annealing and paves the way toward a systematic understanding of this relatively unexplored protocol in a broader context.