Statistical Mechanics of the Quantum K-Satisfiability problem

TL;DR

This paper investigates the quantum K-Satisfiability problem using statistical mechanics, revealing how small quantum fluctuations smooth out classical phase transitions and impact quantum optimization algorithms.

Contribution

It derives a replica-symmetric free energy functional for the quantum K-Satisfiability problem and analyzes the effects of quantum fluctuations on phase transitions.

Findings

Quantum fluctuations eliminate the classical phase transition.

The transition becomes a smooth crossover for small transverse fields.

Numerical solutions show the destruction of the classical transition at small b3.

Abstract

We study the quantum version of the random $K$-Satisfiability problem in the presence of the external magnetic field $\Gamma$ applied in the transverse direction. We derive the replica-symmetric free energy functional within static approximation and the saddle-point equation for the order parameter: the distribution $P[h(m)]$ of functions of magnetizations. The order parameter is interpreted as the histogram of probability distributions of individual magnetizations. In the limit of zero temperature and small transverse fields, to leading order in $\Gamma$ magnetizations $m \approx 0$ become relevant in addition to purely classical values of $m \approx \pm 1$. Self-consistency equations for the order parameter are solved numerically using Quasi Monte Carlo method for K=3. It is shown that for an arbitrarily small $\Gamma$ quantum fluctuations destroy the phase transition present in the classical limit $\Gamma=0$, replacing it with a smooth crossover transition. The implications of this result with respect to the expected performance of quantum optimization algorithms via adiabatic evolution are discussed. The replica-symmetric solution of the classical random $K$-Satisfiability problem is briefly revisited. It is shown that the phase transition at T=0 predicted by the replica-symmetric theory is of continuous type with atypical critical exponents.