Dismantling the Stoquastic Dichotomy

2026-07-20

AI Summary

This work challenges the conventional view that the boundary between classically simulable and quantumly hard Hamiltonians hinges on whether a Hamiltonian is stoquastic, proposing instead that the vanishing geometric phase (VGP) constitutes a more fundamental criterion. The authors construct a 3-local Hamiltonian exhibiting VGP that cannot be made stoquastic via any local basis transformation. They reinterpret the complexity class inclusions MA ⊆ StoqMA ⊆ QMA as demarcating problems with and without VGP. Their analysis shows that the local Hamiltonian problem under VGP is StoqMA-complete, while its frustration-free variant lies in MA. Although VGP can be recognized in polynomial time under natural conditions, determining VGP for general geometrically local Hamiltonians is PSPACE-complete. These results establish non-VGP as a necessary condition for achieving adiabatic quantum advantage.

Abstract

We challenge the notion that a stoquastic binary governs fundamental computational boundaries in quantum computing and classical simulation of quantum systems. We argue that vanishing geometric phase (VGP), a geometric condition on the Hamiltonian's transition graph, more adequately captures these boundaries. To distinguish VGP from stoquasticity, we construct VGP 3-local Hamiltonians that are formally hard to stoquastize, yet belong to a family admitting polynomial-time recognition of the VGP property. Without constructing a stoquastizing unitary, we prove that the local Hamiltonian problem is $\mathsf{StoqMA}$-complete under the promise that the input Hamiltonian has VGP, and that a frustration-free variant is in $\mathsf{MA}$ under the same promise. We use this result to argue that non-VGP is necessary for any claimed adiabatic advantage justified by escaping the $\mathsf{StoqMA}$ regime. Further, we identify natural settings where the VGP property can be recognized in polynomial time. In contrast, we show that recognition of VGP is $\mathsf{PSPACE}$-complete in general for geometrically local Hamiltonians. Our results show that the computational boundaries $\mathsf{MA} \subseteq \mathsf{StoqMA} \subseteq \mathsf{QMA}$ traditionally attributed to stoquasticity are better understood as boundaries between vanishing and non-vanishing geometric phase structure.

Problem

• stoquasticity
• vanishing geometric phase
• computational complexity
• local Hamiltonian problem
• quantum simulation

Innovation

• Vanishing Geometric Phase
• Stoquasticity
• Local Hamiltonian Problem
• StoqMA
• Quantum Computational Complexity