Emulating Majorana fermions and their braiding by Ising spin chains

Abstract

We analyse the control of Majorana zero-energy states by mapping the fermionic system onto a chain of Ising spins. Although the topological protection is lost for the Ising system, the mapping provides additional insight into the nature of the quantum states. By controlling the local magnetic field, one can separate the Ising chain into ferromagnetic and paramagnetic phases, corresponding to topological and non-topological sections of the fermionic system. In this paper we propose (topologically non-protected) protocols performing the braiding operation, and in fact also more general rotations. We first consider a T-junction geometry, but we also propose a protocol for a purely one-dimensional system. Both setups rely on an extra spin-1/2 coupler. By including the extra spin in the T-junction geometry, we overcome limitations due to the 1D character of the Jordan-Wigner transformation. In the 1D geometry the coupler, which controls one of the Ising links, should be manipulated once the ferromagnetic (topological) section of the chain is moved far away. We also propose experimental implementations of our scheme. One is based on a chain of flux qubits which allows for all needed control fields. We also describe how to translate our scheme for the 1D setup to a chain of superconducting wires hosting each a pair of Majorana edge states.

arXiv comment11 pages, 5 figures, version published in PRB

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• Observing and braiding topological Majorana modes on programmable quantum simulators
• Operating in a deep underground facility improves the locking of gradiometric fluxonium qubits at the sweet spots
• Jordan-Wigner transformations for tree structures
• Majorana braiding gates for topological superconductors in a one dimensional geometry
• Fluxon-Based Quantum Simulation in Circuit QED
• Decoherence and relaxation of topological states in extended quantum Ising models
• Coherent multiple-period states in a resonantly driven qubit
• Theory of topological spin Josephson junctions
• Majorana modes in trapped-ion system and their Floquet engineering
• Excitation dynamics in inductively coupled fluxonium circuits
• Probing Majorana Modes via Local Spin Dynamics
• Investigating the Exchange of Ising Chains on a Digital Quantum Computer
• Universal quantum operation of spin-3/2 Blume-Capel chains