Easing the Monte Carlo sign problem

Quantum Monte Carlo (QMC) methods are the gold standard for studying equilibrium properties of quantum many-body systems -- their phase transitions, their ground and thermal state properties. However, in many interesting situations QMC methods are faced with a sign problem, causing the severe limitation of an exponential increase in the sampling complexity and hence the runtime of the QMC algorithm. In this work, we propose and explore a systematic and generally applicable methodology towards alleviating the sign problem by local basis changes, realizing that it is a basis-dependent property. Going significantly beyond previous work on exactly curing the sign problem and model-specific approaches, we introduce the optimization problem of finding the efficiently computable basis in which the sign problem is smallest and refer to this problem as easing the sign problem. We introduce and discuss efficiently computable measures of the severity of the sign problem, and demonstrate that those measures can practically be brought to a good use to ease the sign problem by performing proof-of-principle numerical experiments. Complementing this pragmatic mindset, we prove that easing the sign problem in terms of those measures is in general a computationally hard task for nearest-neighbour Hamiltonians and simple basis choices. Ironically, this holds true even in situations in which finding an exact solution or deciding if such a solution exists is easy.