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A complete list of which models allow this, and whether clever changes of variables can remove the problem in others, is not known.","posed_since":"","precise":"Known sign-free classes of determinant quantum Monte Carlo include models with an antiunitary (Kramers-type) symmetry of the decoupled fermion matrix and models with Majorana time-reversal symmetry (2015 to 2016). Determine necessary and sufficient conditions on a local fermion Hamiltonian for the existence of an auxiliary-field decoupling or local basis change that makes all weights nonnegative, and the computational complexity of deciding it. An answer is a classification theorem or a hardness result for natural Hamiltonian classes.","problem_ref":null,"references":"","settled_by":"A proof giving necessary and sufficient conditions, or a reduction proving the decision problem NP-hard for local fermion Hamiltonians.","status_note":"Results from 2019 to 2020 showed that finding a sign-curing local basis is NP-hard for some qubit Hamiltonian classes; a fermionic classification is incomplete.","title":"Complete criteria for sign-free fermion quantum Monte 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