Mean-Field Solution of the Weak-Strong Cluster Problem for Quantum Annealing with Stoquastic and Non-Stoquastic Catalysts

TL;DR

This paper provides an exact mean-field analysis of the weak-strong cluster problem in quantum annealing, demonstrating how different types of $XX$ interactions and inhomogeneous driving influence the removal of first-order phase transitions.

Contribution

It offers an analytical solution in the thermodynamic limit and clarifies how the placement of $XX$ interactions affects phase transition removal in quantum annealing.

Findings

Non-stoquastic $XX$ interactions can remove first-order transitions.

Placement of $XX$ interactions determines transition removal.

Inhomogeneous transverse field can eliminate first-order transitions.

Abstract

We study the weak-strong cluster problem for quantum annealing in its mean-field version as proposed by Albash [Phys. Rev. A 99 (2019) 042334] who showed by numerical diagonalization that non-stoquastic $XX$ interactions (non-stoquastic catalysts) remove the problematic first-order phase transition. We solve the problem exactly in the thermodynamic limit by analytical methods and show that the removal of the first-order transition is successfully achieved either by stoquastic or non-stoquastic $XX$ interactions depending on whether the $XX$ interactions are introduced within the weak cluster, within the strong cluster, or between them. We also investigate the case where the interactions between the two clusters are sparse, i.e. not of the mean-field all-to-all type. The results again depend on where to introduce the $XX$ interactions. We further analyze how inhomogeneous driving of the transverse field affects the performance of the system without $XX$ interactions and find that inhomogeneity in the transverse field removes the first-order transition if appropriately implemented.