Topological phases and quantum computation

Abstract

This is a collection of lecture notes from three lectures given by Alexei Kitaev at the 2008 Les Houches summer school "Exact methods in low-dimensional physics and quantum computing." They provide a pedagogical introduction to topological phenomena in 1-D superconductors and in the 2-D topological phases of the toric code and honeycomb model.

31 pages. Lectures given by Alexei Kitaev at the 2008 Les Houches Summer School "Exact methods in low-dimensional physics and quantum computing"

References in corpus (6)

• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Quasi-adiabatic Continuation of Quantum States: The Stability of Topological Ground State Degeneracy and Emergent Gauge Invariance
• The Flux-Phase of the Half-Filled Band
• An exact chiral spin liquid with non-Abelian anyons
• Topological multicritical point in the Toric Code and 3D gauge Higgs Models

Cited by in corpus (51)

• Models and Materials for Generalized Kitaev Magnetism
• Physics of the Kitaev model: fractionalization, dynamical correlations, and material connections
• Machine Learning Topological States
• Hilbert Space Fragmentation and Commutant Algebras
• Standard Model Physics and the Digital Quantum Revolution: Thoughts about the Interface
• Quantum phase transition between cluster and antiferromagnetic states
• Topological order in 1D Cluster state protected by symmetry
• Simulating the exchange of Majorana zero modes with a photonic system
• Resource quality of a symmetry-protected topologically ordered phase for quantum computation
• Quantum circuit complexity of one-dimensional topological phases
• Dynamical delocalization of Majorana edge states by sweeping across a quantum critical point
• Superconducting Quantum Simulator for Topological Order and the Toric Code
• Topological Order, Quantum Codes and Quantum Computation on Fractal Geometries
• Locality-Preserving Logical Operators in Topological Stabiliser Codes
• Detection and characterization of symmetry-broken long-range orders in the spin-$\frac{1}{2}$ triangular Heisenberg model
• Complete homotopy invariants for translation invariant symmetric quantum walks on a chain
• Universal quantum computation with ordered spin-chain networks
• Generating topological order from a 2D cluster state using a duality mapping
• Ground state connectivity of local Hamiltonians
• The XYZ${}^2$ hexagonal stabilizer code
• Phase diagram of an interacting staggered Su-Schrieffer-Heeger two-chain ladder close to a quantum critical point
• Fracton physics of spatially extended excitations. II. Polynomial ground state degeneracy of exactly solvable models
• Measurement-Based Quantum Computing with Valence-Bond-Solids
• Superconducting quantum many-body circuits for quantum simulation and computing
• Bosonization based on Clifford algebras and its gauge theoretic interpretation
• Evolution of Dynamical Signature in the X-cube Fracton Topological Order
• Detuning the Honeycomb of the α-RuCl3 Kitaev lattice: A case of Cr3+ dopant
• Ground-State Spaces of Frustration-Free Hamiltonians
• Quantum walks: Schur functions meet symmetry protected topological phases
• Factorized Three-body S-Matrix Restrained by Yang-Baxter Equation and Quantum Entanglements
• Strong-Weak Duality via Jordan-Wigner Transformation: Using Fermionic Methods for Strongly Correlated $su(2)$ Spin Systems
• Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted $BF$ Theories in Five Dimensions
• Measurement-Based Time Evolution for Quantum Simulation of Fermionic Systems
• Finite temperature geometric properties of the Kitaev honeycomb model
• Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors
• Spin, orbital and topological order in models of strongly correlated electrons
• Detection of quantum phase boundary at finite temperatures in integrable spin models
• Quantum Ising dynamics and Majorana-like edge modes in the Rabi lattice model
• Detecting Topological phase transitions in a double kicked quantum rotor
• Disconnected entanglement entropy as a marker of edge modes in a periodically driven Kitaev chain
• Anomalous relaxation and multiply time scales in the quantum $XY$ model with boundary dissipation
• Digital quantum simulation of lattice fermion theories with local encoding
• Recurrent neural network wave functions for Rydberg atom arrays on kagome lattice
• Entanglement in topological systems
• Signature of quantum phase transition manifested in quantum fidelity at finite temperature
• Quantum State Transfer in a Two-dimensional Regular Spin Lattice of Triangular Shape
• Signatures of quantum phases in a dissipative system
• An Eigenvalue Problem for a Fermi System and Lie Algebras
• Dimer XXZ Spin Ladders: Phase diagram and a Non-Trivial Antiferromagnetic Phase
• Emergent Kitaev materials in synthetic Fermi-Hubbard bilayers
• Efficient inference in the transverse field Ising model