Energy gaps in quantum first-order mean-field-like transitions: The problems that quantum annealing cannot solve

TL;DR

This paper investigates the limitations of quantum annealing in solving certain first-order quantum phase transitions by analyzing exactly solvable models and their energy gap closures, revealing fundamental challenges.

Contribution

It provides an exact analysis of energy gaps in mean-field-like quantum phase transitions and links these to the Grover problem, highlighting quantum annealing's limitations.

Findings

Energy gaps close exponentially at first-order transitions.

Quantum annealing struggles with problems exhibiting exponential gap closures.

Connections to the Grover problem illustrate fundamental computational limits.

Abstract

We study first-order quantum phase transitions in models where the mean-field traitment is exact, and the exponentially fast closure of the energy gap with the system size at the transition. We consider exactly solvable ferromagnetic models, and show that they reduce to the Grover problem in a particular limit. We compute the coefficient in the exponential closure of the gap using an instantonic approach, and discuss the (dire) consequences for quantum annealing.