Intrinsic sign problem
For a specified class of local Hamiltonians and allowed local or quasilocal basis changes, characterize obstructions to stoquasticity. Separate finite-input basis-curing complexity from a phase-invariant obstruction under uniformly local transformations.
Problem statement
For a specified class of local Hamiltonians and allowed local or quasilocal basis changes, characterize obstructions to stoquasticity. Separate finite-input basis-curing complexity from a phase-invariant obstruction under uniformly local transformations.
Lean 4
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Background
The sign problem is the exponential obstruction that blocks quantum Monte Carlo for many fermionic and frustrated systems. A Hamiltonian is 'stoquastic' (sign-problem-free in a given basis) if its off-diagonal matrix elements are nonpositive. The question is which systems are only 'basis-dependently' cursed — curable by a local basis change — versus intrinsically sign-problematic.
What is known: Checking the signs of an explicitly listed matrix is easy; a succinct local-Hamiltonian representation and allowed transformations require separate complexity analysis. Hastings, Ringel–Kovrizhin, Smith–Golan–Ringel and Golan–Smith–Ringel establish topological sign-problem obstructions under particular locality, phase and spectral assumptions. Their criteria do not say that every nonzero chiral central charge forbids every conceivable sign-free representation. The result is not supplied by the unrelated digital-simulation paper previously attributed here.
New progress (2026): Karakashian and Hen separated stoquastizability from vanishing geometric phase (VGP). They exhibit VGP 3-local Hamiltonians which are formally hard to stoquastize yet retain StoqMA-type complexity, give polynomial-time recognition in several natural settings, and prove general VGP recognition PSPACE-complete for geometrically local Hamiltonians. The result suggests that the tractable/sign-free boundary is not captured by the existence of a stoquastizing basis alone; it does not provide the sought necessary-and-sufficient quasilocal basis-change criterion.
A general characterization — necessary and sufficient conditions for a local Hamiltonian to be curable by a quasilocal basis change — is open and being extended phase by phase. Related: A9, B10.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
preprint: VGP/stoquastizability and recognition-complexity results concern specified representations, not a complete phase-level sign criterion. Evidence
- [1]Hastings, 'How quantum are non-negative wavefunctions?', J. Math. Phys. 57, 015210 (2016) link
- [2]Smith, Golan & Ringel, intrinsic sign problems in topological quantum matter link
- [3]Golan, Smith & Ringel, intrinsic sign problems in chiral topological matter link
- [4]Karakashian & Hen, 'Dismantling the Stoquastic Dichotomy' (2026 preprint) link
Recorded progress
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Discussion
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