Most Two-Dimensional Bosonic Topological Orders Forbid Sign-Problem-Free Quantum Monte Carlo Simulation: Nonpositive Gauss Sum as an Indicator

Quantum Monte Carlo is a powerful tool for studying quantum many-body physics, yet its efficacy is often curtailed by the notorious sign problem. In this Letter, we introduce a novel criterion for the "intrinsic" sign problem in two-dimensional bosonic topological orders, which cannot be resolved by local basis transformations or adiabatic deformations of the Hamiltonian. Specifically, we find that the positivity of higher Gauss sums is a necessary condition for a two-dimensional bosonic topological order to be realized by a stoquastic Hamiltonian, and hence "sign-problem-free." Equivalently, a nonpositive higher Gauss sum for a given topological order indicates the presence of an intrinsic sign problem. This condition not only aligns with prior findings but significantly broadens their scope. Using this new criterion, we examine the Gauss sums of all 405 bosonic topological orders classified up to rank 12, and strikingly find that 398 of them exhibit intrinsic sign problems. We also uncover intriguing links between the intrinsic sign problem, the gappability of boundary theories, and time-reversal symmetry, suggesting that sign-problem-free quantum Monte Carlo simulation may fundamentally rely on both time-reversal symmetry and gapped boundaries. These results highlight the deep connection between the intrinsic sign problem and fundamental properties of topological phases, offering valuable insights into their classical simulability.