StoqMA versus MA and error amplification

Stoquastic Hamiltonians have nonpositive off-diagonal matrix elements in a specified basis. Their ground-energy complexity motivates StoqMA, introduced by Bravyi–Bessen–Terhal following the earlier stoquastic-Hamiltonian work of Bravyi–DiVincenzo–Oliveira–Terhal. The known containments include $\mathsf{MA}\subseteq\mathsf{StoqMA}\subseteq\mathsf{QMA}\cap\mathsf{SBP}$ and $\mathsf{SBP}\subseteq\mathsf{AM}$. Absence of a sign problem is not a general efficient-simulation theorem.

What is known: Aharonov–Grilo–Liu show that suitable general StoqMA error reduction would imply equality with MA. Liu proves soundness-reduction and other restricted results, but the general completeness-amplification obstacle remains. This is a sharp class-comparison question.

July 2026 preprint: Karakashian–Hen propose vanishing geometric phase (VGP) as a wider Hamiltonian class. They establish StoqMA-completeness of the VGP local-Hamiltonian problem, MA membership of its frustration-free variant, hard-to-stoquastize examples, and PSPACE-completeness of recognizing VGP in their general geometrically local setting. These results refine the landscape without settling StoqMA = MA or general error amplification. Related: B11.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.

preprint: VGP Hamiltonian complexity and recognition results; no resolution of StoqMA versus MA. Evidence