The effect of quantum fluctuations on the coloring of random graphs

TL;DR

This paper investigates how quantum fluctuations influence the coloring problem on random regular graphs, revealing phase transition behaviors and implications for quantum algorithms' efficiency.

Contribution

It applies the quantum cavity method and Monte Carlo simulations to analyze quantum effects on graph coloring, highlighting the failure of quantum adiabatic algorithms.

Findings

Identifies the order of quantum phase transition at low temperatures.

Characterizes the structure of the quantum spin glass phase.

Shows quantum adiabatic algorithms are inefficient due to avoided level crossings.

Abstract

We present a study of the coloring problem (antiferromagnetic Potts model) of random regular graphs, submitted to quantum fluctuations induced by a transverse field, using the quantum cavity method and quantum Monte-Carlo simulations. We determine the order of the quantum phase transition encountered at low temperature as a function of the transverse field and discuss the structure of the quantum spin glass phase. In particular, we conclude that the quantum adiabatic algorithm would fail to solve efficiently typical instances of these problems because of avoided level crossings within the quantum spin glass phase, caused by a competition between energetic and entropic effects.