Abstract
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 sub-class 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, \textit{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 realised 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 offer evidence that it applies more generally with analogous results for non-abelian and chiral theories.
Highlights
Our theoretical and practical understanding of quantum systems involving many interacting particles often relies on our ability to simulate them efficiently
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
Our results extend far beyond previously established results on intrinsic sign problems in two key aspects: they apply to a much larger set of topological quantum field theory (TQFT), and they apply beyond commuting projector Hamiltonians, thereby establishing a direct relation between physical properties of the phase and the sign problem
Summary
Our theoretical and practical understanding of quantum systems involving many interacting particles often relies on our ability to simulate them efficiently. We will prove the following result: Let Hbe a stoquastic gapped nonchiral bosonic Hamiltonian in two dimensions with an Abelian TQFT description at low energy, the topological spins form complete sets of roots of unity This provides a simple criterion for diagnosing intrinsic sign problems in topological models. Our results extend far beyond previously established results on intrinsic sign problems in two key aspects: they apply to a much larger set of TQFTs, and they apply beyond commuting projector Hamiltonians, thereby establishing a direct relation between physical properties of the phase and the sign problem These results place fundamental constraints on the phases that can be realized by stoquastic. It is believed that a wide class of symmetry protected topological phases should be free from sign problems [43,44]
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