2026/07/21 by Armen Karakashian, Itay Hen
#quant-ph #cs.CC #physics.comp-ph
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 StoqMA-complete under the promise that the input Hamiltonian has VGP, and that a frustration-free variant is in 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 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 PSPACE-complete in general for geometrically local Hamiltonians. Our results show that the computational boundaries MA ⊆ StoqMA ⊆ QMA traditionally attributed to stoquasticity are better understood as boundaries between vanishing and non-vanishing geometric phase structure.