The quantum adiabatic algorithm and scaling of gaps at first order quantum phase transitions

TL;DR

This paper investigates the scaling behavior of the energy gap at quantum first order transitions, showing that certain models can have algebraically small gaps, impacting the effectiveness of the quantum adiabatic algorithm.

Contribution

It demonstrates that quantum first order transitions can have algebraically small gaps and constructs simple models with exponential gaps, challenging assumptions about QAA limitations.

Findings

A quantum antiferromagnetic Ising chain can have an algebraically small gap at a first order transition.

A classical 1D Hamiltonian can exhibit an exponential gap at a topological quantum first-order transition.

QAA can succeed or fail at first order transitions depending on the model.

Abstract

Motivated by the quantum adiabatic algorithm (QAA), we consider the scaling of the Hamiltonian gap at quantum first order transitions, generally expected to be exponentially small in the size of the system. However, we show that a quantum antiferromagnetic Ising chain in a staggered field can exhibit a first order transition with only an algebraically small gap. In addition, we construct a simple classical translationally invariant one-dimensional Hamiltonian containing nearest-neighbour interactions only, which exhibits an exponential gap at a thermodynamic quantum first-order transition of essentially topological origin. This establishes that (i) the QAA can be successful even across first order transitions but also that (ii) it can fail on exceedingly simple problems readily solved by inspection, or by classical annealing.