Solution of SAT Problems with the Adaptive-Bias Quantum Approximate Optimization Algorithm

TL;DR

This paper introduces an adaptive-bias QAOA that significantly reduces quantum resources needed to solve hard 3-SAT and Max-3-SAT problems on near-term quantum devices, advancing quantum optimization capabilities.

Contribution

The paper presents an improved adaptive-bias QAOA that enhances performance in hard problem regions and proposes an optimization-free version requiring fewer quantum gates.

Findings

ab-QAOA needs fewer levels than standard QAOA for similar accuracy.

The optimization-free ab-QAOA effectively solves hard 3-SAT problems with fewer gates.

Classical optimization can be replaced by local fields in ab-QAOA.

Abstract

The quantum approximate optimization algorithm (QAOA) is a promising method for solving certain classical combinatorial optimization problems on near-term quantum devices. When employing the QAOA to 3-SAT and Max-3-SAT problems, the quantum cost exhibits an easy-hard-easy or easy-hard pattern respectively as the clause density is changed. The quantum resources needed in the hard-region problems are out of reach for current NISQ devices. We show by numerical simulations with up to 14 variables and analytical arguments that the adaptive-bias QAOA (ab-QAOA) greatly improves performance in the hard region of the 3-SAT problems and hard region of the Max-3-SAT problems. For similar accuracy, on average, ab-QAOA needs 3 levels for 10-variable 3-SAT problems as compared to 22 for QAOA. For 10-variable Max-3-SAT problems, the numbers are 7 levels and 62 levels. The improvement comes from a more targeted and more limited generation of entanglement during the evolution. We demonstrate that classical optimization is not strictly necessary in the ab-QAOA since local fields are used to guide the evolution. This leads us to propose an optimization-free ab-QAOA that can solve the hard-region 3-SAT and Max-3-SAT problems effectively with significantly fewer quantum gates as compared to the original ab-QAOA. Our work paves the way for realizing quantum advantages for optimization problems on NISQ devices.