Complete criteria for sign-free fermion quantum Monte Carlo
In plain words
Some special electron models can be simulated without the sign problem thanks to hidden symmetries. A complete list of which models allow this, and whether clever changes of variables can remove the problem in others, is not known.
Precise statement
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.
What would settle it
A proof giving necessary and sufficient conditions, or a reduction proving the decision problem NP-hard for local fermion Hamiltonians.
Status in the literature
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.