Classical and Quantum Annealing in the Median of Three Satisfiability

TL;DR

This paper compares classical and quantum annealing complexities for a specific 3-satisfiability problem ensemble, finding both grow exponentially with problem size and classical methods outperform quantum in this case.

Contribution

It provides the first detailed analysis of classical and quantum complexities for median 3-SAT problems with unique solutions, revealing exponential growth and classical advantage.

Findings

Both classical and quantum complexities diverge exponentially with problem size.

Quantum growth-rate constant is 3.8 times larger than classical, favoring classical fluctuations.

Standard adiabatic quantum computation does not achieve polynomial complexity.

Abstract

We determine the classical and quantum complexities of a specific ensemble of three-satisfiability problems with a unique satisfying assignment for up to N=100 and N=80 variables, respectively. In the classical limit we employ generalized ensemble techniques and measure the time that a Markovian Monte Carlo process spends in searching classical ground states. In the quantum limit we determine the maximum finite correlation length along a quantum adiabatic trajectory determined by the linear sweep of the adiabatic control parameter in the Hamiltonian composed of the problem Hamiltonian and the constant transverse field Hamiltonian. In the median of our ensemble both complexities diverge exponentially with the number of variables. Hence, standard, conventional adiabatic quantum computation fails to reduce the computational complexity to polynomial. Moreover, the growth-rate constant in the quantum limit is 3.8 times as large as the one in the classical limit, making classical fluctuations more beneficial than quantum fluctuations in ground-state searches.