Simulation of topological field theories by quantum computers

Abstract

Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H = 0. These are called topological quantum filed theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.

Cited by in corpus (125)

• Quantum Simulation
• The Physical Implementation of Quantum Computation
• Fermionic quantum computation
• Quantum algorithms for algebraic problems
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Ground state entanglement and geometric entropy in the Kitaev's model
• Physics, Topology, Logic and Computation: A Rosetta Stone
• High-order quantum algorithm for solving linear differential equations
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• Braiding Operators are Universal Quantum Gates
• Superselection rules and quantum protocols
• NP-complete Problems and Physical Reality
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• Demonstrating anyonic fractional statistics with a six-qubit quantum simulator
• Introduction to topological quantum computation with non-Abelian anyons
• Robustness of non-abelian holonomic quantum gates against parametric noise
• Universal Quantum Computation with Hybrid Spin-Majorana Qubits
• The BQP-hardness of approximating the Jones Polynomial
• Fractional quantum Hall effect at $\nu =5/2$: Ground states, non-Abelian quasiholes, and edge modes in a microscopic model
• A hierarchy of topological tensor network states
• Homological Error Correction: Classical and Quantum Codes
• Quantum algorithm for non-homogeneous linear partial differential equations
• Fast-forwarding of Hamiltonians and Exponentially Precise Measurements
• Five Two-Qubit Gates Are Necessary for Implementing Toffoli Gate
• Thermopower as a Possible Probe of Non-Abelian Quasiparticle Statistics in Fractional Quantum Hall Liquids
• How hard is it to approximate the Jones polynomial?
• Spin-dependent coupling between quantum dots and topological quantum wires
• Optical Simulation of Yang-Baxter Equation
• Trading inverses for an irrep in the Solovay-Kitaev theorem
• Quantum Computation and Quantum Information
• Methods for simulating string-net states and anyons on a digital quantum computer
• Edge Excitations and Non-Abelian Statistics in the Moore-Read State: A Numerical Study in the Presence of Coulomb Interaction and Edge Confinement
• Majorana Fermions in Magnetic Chains
• Non-adiabatic geometrical quantum gates in semiconductor quantum dots
• Holonomic quantum gates: A semiconductor-based implementation
• Computing Spin Networks
• Protected gates for topological quantum field theories
• Graph isomorphism and Gaussian boson sampling
• Topological Rényi entropy after a quantum quench
• Scaling of Entanglement Entropy at 2D quantum Lifshitz fixed points and topological fluids
• Quantum Algorithm for Fidelity Estimation
• From Quantum Groups to Unitary Modular Tensor Categories
• Experimental approximation of the Jones polynomial with DQC1
• Level Reduction and the Quantum Threshold Theorem
• On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
• Towards topological quantum computer
• Time Reversal and Charge Conjugation in an Embedding Quantum Simulator
• Thermalization, Error-Correction, and Memory Lifetime for Ising Anyon Systems
• Quantum Computation Beyond the Circuit Model
• Fault-Tolerant Quantum Error Correction for non-Abelian Anyons
• Probing topological order with Rényi entropy
• Efficient quantum circuits for arbitrary sparse unitaries
• Constructing Functional Braids for Low-Leakage Topological Quantum Computing
• Bulk detection of time-dependent topological transitions in quenched chiral models
• Local characterization of 1d topologically ordered states
• Several natural BQP-Complete problems
• On the Quantum Computational Complexity of the Ising Spin Glass Partition Function and of Knot Invariants
• Quantum hashing with the icosahedral group
• Direct Experimental Simulation of the Yang-Baxter Equation
• Localization of unitary braid group representations
• The breaking of quantum double symmetries by defect condensation
• Local response of topological order to an external perturbation
• Quantum Adiabatic Computation With a Constant Gap is Not Useful in One Dimension
• From Topological to Quantum Entanglement
• Mapping of Parent Hamiltonians: from Abelian and non-Abelian Quantum Hall States to Exact Models of Critical Spin Chains
• Grover search and the no-signaling principle
• Chiral Partition Functions of Quantum Hall Droplets
• Adiabatic topological quantum computing
• Nested Topological Order
• All Pure Two-Qudit Entangled States Can be Generated via a Universal Yang--Baxter Matrix Assisted by Local Unitary Transformations
• The Kitaev-Ising model, Transition between topological and ferromagnetic order
• Local Hamiltonians in Quantum Computation
• A comparative study of universal quantum computing models: towards a physical unification
• Circuit Complexity in Topological Quantum Field Theory
• Classical Ising model test for quantum circuits
• Concatenation of the Gottesman-Kitaev-Preskill code with the XZZX surface code
• Fault-tolerant Holonomic Quantum Computation in Surface Codes
• Local convertibility of the ground state of the perturbed Toric code
• Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation
• Trapping Abelian anyons in fractional quantum Hall droplets
• Holographic Software for Quantum Networks
• Error-Resistant Distributed Quantum Computation in Trapped Ion Chain
• A complete characterization of the spectrum of the Kitaev model on spin ladders
• Signatures of Majorana bound states and parity effects in two-dimensional chiral p-wave Josephson junctions
• The Fibonacci Model and the Temperley-Lieb Algebra
• Complexity of Supersymmetric Systems and the Cohomology Problem
• Space-bounded Church-Turing thesis and computational tractability of closed systems
• An Interferometry-Free Protocol for Demonstrating Topological Order
• Circuit Complexity of Knot States in Chern-Simons theory
• Exploiting Geometric Degrees of Freedom in Topological Quantum Computing
• Thermalization of Topological Entropy after a Quantum Quench
• Braid Matrices and Quantum Gates for Ising Anyons Topological Quantum Computation
• Unified approach to topological quantum computation with anyons: From qubit encoding to Toffoli gate
• Information-theoretical formulation of anyonic entanglement
• Quantum Fourier Transforms and the Complexity of Link Invariants for Quantum Doubles of Finite Groups
• Topological Quantum Gate Construction by Iterative Pseudogroup Hashing
• Anyons in Geometric Models of Matter
• Elucidating Dicke Superradiance by quantum uncertainty
• Entanglement Classification from a Topological Perspective
• Modeling free anyons at the bosonic and fermionic ends
• Quantum Simulation of the Two-Dimensional Weyl Equation in a Magnetic Field
• Computational Distinguishability of Quantum Channels
• Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit
• Quantum computation and the evaluation of tensor networks
• Quantum computation from fermionic anyons on a 1D lattice
• Quantum Key Distribution with no Shared Reference Frame
• Magnetization and Spin Excitations of Non-Abelian Quantum Hall States
• Connectomes and Properties of Quantum Entanglement
• Drinfeld centers of fusion categories arising from generalized Haagerup subfactors
• Holonomic swap and controlled-swap gates of neutral atoms via selective Rydberg pumping
• Sumcheck-based delegation of quantum computing to rational server
• A Survey of Quantum Programming Languages: History, Methods, and Tools
• Conformal Field Theories as Scaling Limit of Anyonic Chains
• On a suggestion relating topological and quantum mechanical entanglements
• Field Theoretic Aspects of Condensed Matter Physics: An Overview
• Generalisations of Hecke algebras from Loop Braid Groups
• Knots and signal transmission in topological quantum devices
• Elementary Particles as Gates for Universal Quantum Computation
• Studies on the Role of Entanglement in Mixed-state Quantum Computation
• The Universal Edge Physics in Fractional Quantum Hall Liquids
• A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes
• Quantum Fibrations: quantum computation on an arbitrary topological space
• Topological quantum computation is hyperbolic
• Grobner Bases for Finite-temperature Quantum Computing and their Complexity
• Quantum Computers: Engines for Next Industrial Revolution