Enhancing the Energy Gap of Random Graph Problems via XX-catalysts in Quantum Annealing

TL;DR

This paper demonstrates that using multiple XX-catalysts in quantum annealing significantly increases the energy gap for MWIS problems, especially during severe phase transitions, improving the efficiency of quantum optimization.

Contribution

The study introduces a novel application of multiple XX-catalysts to enhance energy gaps in quantum annealing for MWIS problems, showing effectiveness over non-stoquastic catalysts.

Findings

Multiple XX-catalysts significantly increase the minimum energy gap.

The effectiveness of the catalyst correlates with the severity of the phase transition.

Stoquastic catalysts outperform non-stoquastic versions in this context.

Abstract

One of the bottlenecks in solving combinatorial optimisation problems using quantum annealers is the emergence of exponentially-closing energy gaps between the ground state and the first excited state during the annealing, which indicates that a first-order phase transition is taking place. The minimum energy gap scales inversely with the exponential of the system size, ultimately resulting in an exponentially large time required to ensure the adiabatic evolution. In this paper we demonstrate that employing multiple XX-catalysts on all the edges of a graph upon which a MWIS (Maximum Weighted Independent Set) problem is defined significantly enhances the minimum energy gap. Remarkably, our analysis shows that the more severe the first-order phase transition, the more effective the catalyst is in opening the gap. This result is based on a detailed statistical analysis performed on a large number of randomly generated MWIS problem instances on both Erd\H{o}s-R\'enyi and Barab\'asi-Albert graphs. We also observe that similar performance cannot be achieved by the non-stoquastic version of the same catalyst, with the stoquastic catalyst being the preferred choice in this context.