First-order transitions and the performance of quantum algorithms in random optimization problems

TL;DR

This paper investigates the phase transition in a quantum optimization problem, revealing a first-order transition with an exponentially closing gap, implying exponential time complexity for the Quantum Adiabatic Algorithm.

Contribution

It characterizes the nature of the phase transition in a quantum optimization problem and demonstrates the exponential difficulty faced by quantum algorithms.

Findings

First-order quantum phase transition identified

Exponential gap closing at the transition

Quantum Adiabatic Algorithm requires exponential runtime

Abstract

We present a study of the phase diagram of a random optimization problem in presence of quantum fluctuations. Our main result is the characterization of the nature of the phase transition, which we find to be a first-order quantum phase transition. We provide evidence that the gap vanishes exponentially with the system size at the transition. This indicates that the Quantum Adiabatic Algorithm requires a time growing exponentially with system size to find the ground state of this problem.