Search for non-Abelian Majorana braiding statistics in superconductors

Abstract

This is a tutorial review of methods to braid the world lines of non-Abelian anyons (Majorana zero-modes) in topological superconductors. That "Holy Grail" of topological quantum information processing has not yet been reached in the laboratory, but there now exists a variety of platforms in which one can search for the Majorana braiding statistics. After an introduction to the basic concepts of braiding we discuss how one might be able to braid immobile Majorana zero-modes, bound to the end points of a nanowire, by performing the exchange in parameter space, rather than in real space. We explain how Coulomb interaction can be used to both control and read out the braiding operation, even though Majorana zero-modes are charge neutral. We ask whether the fusion rule might provide for an easier pathway towards the demonstration of non-Abelian statistics. In the final part we discuss an approach to braiding in real space, rather than parameter space, using vortices injected into a chiral Majorana edge mode as "flying qubits". Contents: I. Introduction II. Basic Concepts (The magic of braiding; Non-Abelian statistics; Fusion rules; Clifford gates; Topological protection) III. Braiding of Majorana zero-modes in nanowires (The three-point turn; Non-Abelian Berry phase; Coulomb-assisted braiding; Anyon teleportation) IV. Read-out of Majorana qubits (Majorana interferometry; Inductive coupling to a flux qubit; Microwave coupling to a transmon qubit; Capacitive coupling to a quantum dot; Random Access Majorana Memory) V. Fusion of Majorana zero-modes in nanowires (Linear junction or tri-junction; If we can fuse, do we need to braid?) VI. How to braid Majorana edge modes (Chiral edge modes in a superconductor; Edge vortex injection; Construction of the vortex operator; Edge vortex braiding)

arXiv comment30 pages, 15 figures; Lecture notes for the Les Houches summer school on Quantum Information Machines (July 2019)

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• Helical Liquids in Semiconductors
• Yu-Shiba-Rusinov qubit
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• Steering Majorana braiding via skyrmion-vortex pairs: a scalable platform
• Topological Quantum Computing Using Nanowire Devices
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• Fusion rules in a Majorana single-charge transistor
• Tunable vortex Majorana modes controlled by strain in homogeneous LiFeAs
• Near zero-energy Caroli-de Gennes-Matricon vortex states in the presence of impurities
• Many-body Majorana braiding without an exponential Hilbert space
• Corner states, hinge states and Majorana modes in SnTe nanowires
• Readout of parafermionic states by transport measurements
• Phenomenology of Majorana zero modes in full-shell hybrid nanowires
• Robust topological superconductivity in weakly coupled nanowire-superconductor hybrid structures
• Topological superconductivity enhanced by exceptional points
• Strong nonlocal tuning of the current-phase relation of a quantum dot based Andreev molecule
• Non-Hermitian zero-energy pinning of Andreev and Majorana bound states in superconductor-semiconductor systems
• Demonstrating Majorana non-Abelian properties using fast adiabatic charge-transfer
• Entanglement measures of Majorana bound states
• Quantum-dot-based Kitaev chains: Majorana quality measures and scaling with increasing chain length
• Interplay between Majorana and Shiba states in a minimal Kitaev chain coupled to a superconductor
• Chemical Principles of Intrinsic Topological Superconductors
• One-dimensional non-interacting topological insulators with chiral symmetry
• Non-Abelian Half-Quantum Vortices in ${}^3{P}_2$ Topological Superfluids
• Non-Hermitian minimal Kitaev chains
• Majorana sweet spots in 3-site Kitaev chains
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• Emergent pair symmetries in systems with poor man's Majorana modes
• Non-Abelian Anyons and Non-Abelian Vortices in Topological Superconductors
• Multiterminal transport spectroscopy of subgap states in Coulomb-blockaded superconductors
• Josephson effect and critical currents in trivial and topological full-shell hybrid nanowires
• Quantized spin pumping in topological ferromagnetic-superconducting nanowires
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• Absence of Majorana oscillations in finite-length full-shell hybrid nanowires
• Statistical Majorana Bound State Spectroscopy
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• Tunable topological states hosted by unconventional superconductors with adatoms
• Majorana-Kondo competition in a cross-shaped double quantum dot-topological superconductor system
• Unconventional topological transitions in a self-organized magnetic ladder
• Topological superconductivity in a magnetic-texture coupled Josephson junction
• Topological indicators for systems with open boundaries: Application to the Kitaev wire
• Subgap states in semiconductor-superconductor devices for quantum technologies: Andreev qubits and minimal Majorana chains
• Phase jumps in Josephson junctions with time-dependent spin-orbit coupling
• Interplay of quantum spin Hall effect and spontaneous time-reversal symmetry breaking in electron-hole bilayers II: Zero-field topological superconductivity
• Photonic cross-noise spectroscopy of Majorana bound states
• Probing Majorana bound states via a pn-junction containing a quantum dot
• The topological Kondo model out of equilibrium
• Sub-symmetry Protected Topology in Topological Insulators and Superconductors
• Quantized Thermal Hall Conductance and the Topological Phase Diagram of a Superconducting Bismuth Bilayer
• Probing the non-Abelian fusion of a pair of Majorana zero modes
• Interaction- and phonon-induced topological phase transitions in double helical liquids
• Preparation and readout of Majorana qubits in magnet-superconductor hybrid systems
• Effects of electron-electron interactions in the Yu-Shiba-Rusinov lattice model
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• Engineering subgap states in superconductors by the symmetry of altermagnetism
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