  Although stoquastic Hamiltonians are known to be simulable via
sign-problem-free quantum Monte Carlo (QMC) techniques, the non-stoquasticity
of a Hamiltonian does not necessarily imply the existence of a QMC sign
problem. We give a sufficient and necessary condition for the QMC-simulability
of Hamiltonians in a fixed basis in terms of geometric phases associated with
the chordless cycles of the weighted graphs whose adjacency matrices are the
Hamiltonians. We use our findings to provide a construction for non-stoquastic,
yet sign-problem-free and hence QMC-simulable, quantum many-body models. We
also demonstrate why the simulation of truly sign-problematic models using the
QMC weights of the stoquasticized Hamiltonian is generally sub-optimal. We
offer a superior alternative.
