Emulating Majorana fermions and their braiding by Ising spin chains

We analyze 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 nontopological sections of the fermionic system. In this paper we propose (topologically nonprotected) 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 (1D) system. Both setups rely on an extra spin-$\frac{1}{2}$ 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.

Figure 1

Three Ising chains. (left panel) Ising coupling between the chains. Here, a fictitious spin $\vec{S}$ is introduced formally in order to construct the Klein factors (see text). (right panel) A central spin $\vec{S}$ controls the coupling between the chains via a three-spin interaction.

Figure 2

Two Ising chains coupled via the central spin $\vec{S}$.

Figure 3

Practical realization of qubits chains with Josephson junction circuits. (a) Basic states of the gradiometer flux qubit. A permanent flux of half a flux quantum is applied to each loop. Tunnelling occurs through one smaller junction denoted by $\alpha$. (b) Driving of the qubit by changing the tunneling strength (${\sigma }^x$) or the flux bias (${\sigma }^z$). (c) Layout of the system with two chains (left and right) and the coupler (center).

Figure 4

Implementation of a Kitaev chain using superconducting islands on top of a semiconducting wire (see Ref. [15]). The gates control both the electrochemical potential of each wire (transverse field in the Ising equivalent) as well as the tunneling amplitude between the adjacent Majoranas (analog of $J$ in the Ising chain).

Figure 5

A flux qubit controls one of the links in the Kitaev chain. Assuming the Josephson energy of the qubit dominates over the tunneling amplitude of Majoranas, the latter is enslaved to the quantum state of the qubit.