On quantum mean-field models and their quantum annealing

TL;DR

This paper analyzes quantum mean-field spin models with p-body interactions, focusing on phase transitions, spectral gaps, and annealing dynamics to evaluate quantum adiabatic algorithms' efficiency.

Contribution

It provides a detailed analytical study of static and dynamic properties of p-spin quantum models, including gap decay and annealing effects, especially at first-order transitions.

Findings

Exponential decay rate of spectral gap at first-order transition derived

Impact of first-order transition on residual excitation energy analyzed

Differences between p=2 and p>2 models in phase transition behavior identified

Abstract

This paper deals with fully-connected mean-field models of quantum spins with p-body ferromagnetic interactions and a transverse field. For p=2 this corresponds to the quantum Curie-Weiss model (a special case of the Lipkin-Meshkov-Glick model) which exhibits a second-order phase transition, while for p>2 the transition is first order. We provide a refined analytical description both of the static and of the dynamic properties of these models. In particular we obtain analytically the exponential rate of decay of the gap at the first-order transition. We also study the slow annealing from the pure transverse field to the pure ferromagnet (and vice versa) and discuss the effect of the first-order transition and of the spinodal limit of metastability on the residual excitation energy, both for finite and exponentially divergent annealing times. In the quantum computation perspective this quantity would assess the efficiency of the quantum adiabatic procedure as an approximation algorithm.