Mean Field Analysis of Quantum Annealing Correction

TL;DR

This paper uses mean-field analysis to understand how quantum annealing correction (QAC) improves error suppression in quantum annealers, revealing its effects on phase transitions and energy gaps in models with and without disorder.

Contribution

It provides an analytical mean-field framework to explain the mechanisms of QAC in different quantum spin models, including the effects on phase transitions and energy gaps.

Findings

QAC shifts the phase transition to higher transverse fields for p=2.

QAC prevents the closing of the energy gap for large penalties when p≥3.

Results are consistent in models with and without disorder, like the Hopfield model.

Abstract

Quantum annealing correction (QAC) is a method that combines encoding with energy penalties and decoding to suppress and correct errors that degrade the performance of quantum annealers in solving optimization problems. While QAC has been experimentally demonstrated to successfully error-correct a range of optimization problems, a clear understanding of its operating mechanism has been lacking. Here we bridge this gap using tools from quantum statistical mechanics. We study analytically tractable models using a mean-field analysis, specifically the $p$-body ferromagnetic infinite-range transverse-field Ising model as well as the quantum Hopfield model. We demonstrate that for $p=2$, where the phase transition is of second order, QAC pushes the transition to increasingly larger transverse field strengths. For $p\ge3$, where the phase transition is of first order, QAC softens the closing of the gap for small energy penalty values and prevents its closure for sufficiently large energy penalty values. Thus QAC provides protection from excitations that occur near the quantum critical point. We find similar results for the Hopfield model, thus demonstrating that our conclusions hold in the presence of disorder.