Intrinsic sign problems in topological quantum field theories

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.

arXiv comment11 pages, 4 figures (+6 pages appendices, 3 figures) published version

References in corpus (44)

• Fault-tolerant quantum computation by anyons
• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Topological entanglement entropy
• Quantum Spin Liquids
• String-net condensation: A physical mechanism for topological phases
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Spin Liquid Ground State of the $S=1/2$ Kagome Heisenberg Model
• Characterizing Quantum Supremacy in Near-Term Devices
• Zoo of quantum-topological phases of matter
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• A Field Guide to Spin Liquids
• Fractional statistics in anyon collisions
• Machine learning quantum phases of matter beyond the fermion sign problem
• Quasi-particle Statistics and Braiding from Ground State Entanglement
• Emergent statistical mechanics of entanglement in random unitary circuits
• Self-Learning Monte Carlo Method
• Quantum Hall Physics - hierarchies and CFT techniques
• Signatures of Dirac cones in a DMRG study of the Kagome Heisenberg model
• A Short Introduction to Topological Quantum Computation
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
• Characterizing topological order by studying the ground states of an infinite cylinder
• Gapped spin liquid with ${\mathbb{Z}}_2$-topological order for kagome Heisenberg model
• Quantum computation with Turaev-Viro codes
• Generalizations and limitations of string-net models
• Monte Carlo Studies of Quantum Critical Metals
• Easing the Monte Carlo sign problem
• An invariant of topologically ordered states under local unitary transformations
• On the Computational Complexity of Curing the Sign Problem
• Modular Matrices as Topological Order Parameter by Gauge Symmetry Preserved Tensor Renormalization Approach
• Universal Topological Data for Gapped Quantum Liquids in Three Dimensions and Fusion Algebra for Non-Abelian String Excitations
• Protected gates for topological quantum field theories
• Quantum gravity kinematics from extended TQFTs
• Minimal Entangled States and Modular Matrix for Fractional Quantum Hall Effect in Topological Flat Bands
• Universal Wave Function Overlap and Universal Topological Data from Generic Gapped Ground States
• Exact realization of Integer and Fractional Quantum Hall Phases in U(1)xU(1) models in (2+1)d
• How Quantum Are Non-Negative Wavefunctions?
• Quantum Origami: Transversal Gates for Quantum Computation and Measurement of Topological Order
• Topological Origin of the Fermion Sign Problem
• Modular matrices from universal wave function overlaps in Gutzwiller-projected parton wave functions
• Instantaneous braids and Dehn twists in topologically ordered states
• Classical topological paramagnetism
• A bosonic topological phase in a paired superfluid

Cited by in corpus (21)

• Probing sign structure using measurement-induced entanglement
• Symmetry-protected sign problem and magic in quantum phases of matter
• Evidence for deconfined $U(1)$ gauge theory at the transition between toric code and double semion
• Quantum Simulation of the Bosonic Kitaev Chain
• Expressive power of complex-valued restricted Boltzmann machines for solving non-stoquastic Hamiltonians
• Building models of topological quantum criticality from pivot Hamiltonians
• Intrinsic sign problem in fermionic and bosonic chiral topological matter
• Fermion sign bounds theory in quantum Monte Carlo simulation
• Sign Problem in Quantum Monte Carlo Simulation
• Non-Hermitian effects of the intrinsic signs in topologically ordered wavefunctions
• Mitigating the fermion sign problem by automatic differentiation
• Robust Fermi liquid instabilities in sign problem-free models
• Defining a universal sign to strictly probe a phase transition
• Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat's Lemma
• Continuous topological phase transition between ${\mathbb{Z}}_2$ topologically ordered phases
• Constraints from anti-unitary symmetries on phase diagrams of sign problem-free models
• Entanglement-enabled symmetry-breaking orders
• Most two-dimensional bosonic topological orders forbid sign-problem-free quantum Monte Carlo: Nonpositive Gauss sum as an indicator
• Twisted quantum doubles are sign problem-free
• Complex path simulations of geometrically frustrated ladders
• Quantum spin liquids bootstrapped from Ising criticality in Rydberg arrays