  The infamous sign problem leads to an exponential complexity in Monte Carlo
simulations of generic many-body quantum systems. Nevertheless, many phases of
matter are known to admit a sign-problem-free representative, allowing
efficient simulations on classical computers. Motivated by long standing open
problems in many-body physics, as well as fundamental questions in quantum
complexity, the possibility of intrinsic sign problems, where a phase of matter
admits no sign-problem-free representative, was recently raised but remains
largely unexplored. Here, we establish the existence of an intrinsic sign
problem in a broad class of gapped, chiral, topological phases of matter.
Within this class, we exclude the possibility of stoquastic Hamiltonians for
bosons (or 'qudits'), and of sign-problem-free determinantal Monte Carlo
algorithms for fermions. The intrinsically sign-problematic class of phases we
identify is defined in terms of topological invariants with clear observable
signatures: the chiral central charge, and the topological spins of anyons. We
obtain analogous results for phases that are spontaneously chiral, and present
evidence for an extension of our results that applies to both chiral and
non-chiral topological matter.
