Most two-dimensional bosonic topological orders forbid sign-problem-free quantum Monte Carlo: Nonpositive Gauss sum as an indicator

TL;DR

This work provides a rigorous criterion for the intrinsic sign problem in two-dimensional bosonic topological orders, showing that all higher Gauss sums must be positive for a sign-free (stoquastic) realization to exist. By linking $\tau_n = \sum_a d_a^2 \theta_a^n$ positivity to the positivity of a partial-rotation amplitude on a cylinder, the authors connect topological data to classical simulability and derive constraints on edge gappability and time-reversal symmetry. Applying the criterion to a comprehensive classification of 405 bosonic topological orders up to rank 12, they find that 398 orders exhibit intrinsic sign problems, with only a few proven sign-free (notably the toric code, $S_3$ and $\mathbb{Z}_3$ gauge theories). The results illuminate deep connections between intrinsic sign problems, gapped boundaries, and symmetry, and point to fundamental limits on classical simulation for most bosonic topological phases, while outlining future directions for refining the diagnostic and extending to higher dimensions.

Abstract

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, gappability of boundary theories, and time-reversal symmetry, suggesting that sign-problem-free quantum Monte Carlo 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.

Figures (2)

• Figure 1: (a) Lattice system on a cylinder. The linear system size is sufficiently longer than the correlation length. Each lattice site is associated with a local Hilbert space $\mathcal{H}_i \simeq \mathbb{C}^d$. The cylinder is partitioned into two subregions, $L$ and $R$. (b) Partial rotation by an angle $\varphi = 2\pi / n$ on the subregion $R$. (c) Venn diagram illustrating the relationship between our sign-free condition $\tau_n > 0$ and the intrinsic sign problem of two-dimensional bosonic topological orders. The grey and white regions represent topological orders exhibiting the intrinsic sign problem and sign-free topological orders, respectively. Sign-free topological orders necessarily satisfy our sign-free condition $\tau_n > 0$. The set of topological orders whose intrinsic sign problems are undecidable is represented by the shaded region. Among 405 topological orders up to rank 12 ng2023classification, we find 398 cases suffering the intrinsic sign problem. Only 3 topological orders up to rank 12 are sign-free. The intrinsic sign problems of the other 4 are undecidable. (d) Venn diagram illustrating the relationship between time-reversal symmetry, gapped boundaries, and sign-free Abelian topological orders. Every sign-free Abelian topological order is time-reversal symmetric and admits a gapped boundary. For non-Abelian topological orders, we derive similar, though slightly weaker, conclusions. Please refer to the main text for details.
• Figure 2: Illustration for the construction of the ground state in the superselection sector labeled by an anyon $a$. Starting with the trivial superselection sector, denoted by $\mathds{1}$, we first create a particle-antiparticle pair in the bulk. The anyon $a$ and its anti-anyon $\bar{a}$ are adiabatically dragged toward each end of the cylinder. When $a$ and $\bar{a}$ move to the spatial infinity, the state is now threaded the anyon flux $a$ through the cylinder and in the superselection sector labeled by $a$.

Theorems & Definitions (1)

• Conjecture 1