Quantum annealing for hard 2-SAT problems : Distribution and scaling of minimum energy gap and success probability

TL;DR

This paper investigates the scaling behavior of quantum annealing on hard 2-SAT problems, analyzing how modifications to the Hamiltonian affect success probability and comparing quantum and simulated annealing performance.

Contribution

It introduces trigger Hamiltonians to quantum annealing, demonstrating improved scaling constants and potential advantages over simulated annealing for certain configurations.

Findings

Scaling of run-time is exponential for all Hamiltonians.

Trigger Hamiltonians can reduce the scaling constant significantly.

Quantum annealers show competitive performance compared to simulations.

Abstract

In recent years, quantum annealing has gained the status of being a promising candidate for solving various optimization problems. Using a set of hard 2-satisfiabilty (2-SAT) problems, consisting of upto 18-variables problems, we analyze the scaling complexity of the quantum annealing algorithm and study the distributions of the minimum energy gap and the success probability. We extend the analysis of the standard quantum annealing Hamiltonian by introducing an additional term, the trigger Hamiltonian, which can be of two types : ferromagnetic and antiferromagnetic. We use these trigger Hamiltonians to study their influence on the success probability for solving the selected 2-SAT problems. We found that although the scaling of the run-time is exponential for the standard and modified quantum annealing Hamiltonians, the scaling constant in case of adding the trigger Hamiltonians can be significantly smaller. Furthermore, certain choices for the trigger Hamiltonian and annealing times can result in a better scaling than that for simulated annealing. Lastly, we also use the quantum annealers of D-Wave Systems Inc. to study their performance in solving the 2-SAT problems and compare it with the simulation results.