Intrinsic sign problems in topological quantum field theories

The sign problem is a widespread numerical hurdle preventing us from simulating the equilibrium behavior of various problems at the forefront of physics. Focusing on an important subclass of such problems, bosonic (2+1)-dimensional topological quantum field theories, here we provide a simple criterion to diagnose intrinsic sign problems-that is, sign problems that are inherent to that phase of matter and cannot be removed by any local unitary transformation. Explicitly, if the exchange statistics of the anyonic excitations do not form complete sets of roots of unity, then the model has an intrinsic sign problem. This establishes a concrete connection between the statistics of anyons, contained in the modular S and T matrices, and the presence of a sign problem in a microscopic Hamiltonian. Furthermore, it places constraints on the phases that can be realized by stoquastic Hamiltonians. We prove this and a more restrictive criterion for the large set of gapped bosonic models described by an Abelian topological quantum field theory at low-energy, and we offer evidence that it applies more generally with analogous results for non-Abelian and chiral theories.