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We study the weak-strong cluster problem for quantum annealing in itsmean-field version as proposed by Albash [Phys. Rev. A 99 (2019) 042334] whoshowed 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 methodsand show that the removal of the first-order transition is successfullyachieved either by stoquastic or non-stoquastic $XX$ interactions depending onwhether the $XX$ interactions are introduced within the weak cluster, withinthe strong cluster, or between them. We also investigate the case where theinteractions between the two clusters are sparse, i.e. not of the mean-fieldall-to-all type. The results again depend on where to introduce the $XX$interactions. We further analyze how inhomogeneous driving of the transversefield affects the performance of the system without $XX$ interactions and findthat inhomogeneity in the transverse field removes the first-order transitionif appropriately implemented.

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