Teleportation-based quantum information processing with Majorana zero modes

Sagar Vijay and Liang Fu

Phys. Rev. B 94, 235446 – Published 29 December 2016

Abstract

We present a measurement-based scheme for performing braiding operations on Majorana zero modes in mesoscopic superconductor islands and for detecting their non-Abelian statistics without moving or hybridizing them. In our scheme for “braiding without braiding”, the topological qubit encoded in any pair of well-separated Majorana zero modes is read out from the transmission phase shift in electron teleportation through the island in the Coulomb-blockade regime. We propose experimental setups to measure the teleportation phase shift via conductance in an electron interferometer or persistent current in a closed loop.

Received 13 September 2016

DOI:https://doi.org/10.1103/PhysRevB.94.235446

Physics Subject Headings (PhySH)

Majorana fermions Quantum information processing

Nanowires

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Sagar Vijay^1,2 and Liang Fu¹

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Kavli Institute for Theoretical Physics, Santa Barbara, California 93106, USA

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Issue

Vol. 94, Iss. 23 — 15 December 2016

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Images

Figure 1
Majorana interferometer. Two electron interferometry setups to measure the topological qubit formed by Majorana zero modes $γ_{1}$ and $γ_{2}$ . In both interferometers, one path goes through the topological qubit while the other path goes through (a) a normal metal with a sufficiently long phase-coherence length (blue) and (b) a second Majorana island initialized in a definite parity state, $i ψ_{1} ψ_{2} = \pm 1$ .
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Figure 2
Majorana SQUID. When the two Majorana zero modes $γ_{1}$ and $γ_{2}$ are connected by a bridge outside the island to form a closed loop, with the bridge being (a) a normal metal with a sufficiently long phase-coherence length or (b) a reference Majorana island in a definite parity state $i ψ_{1} ψ_{2} = \pm 1$ , the topological qubit defined by $i γ_{1} γ_{2} = \pm 1$ may be read out by measuring the persistent current $I$ in the ground state, which is a $h / e$ -periodic function of the applied flux $Φ$ .
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Figure 3
Teleportation phase shift. Braiding (a) or exchanging (b) Majorana zero modes induces a transformation on the wave functions as indicated. The shaded lines shown above are physical regions where the superconducting phase rapidly advances by $2 π$ . The sign change in the transmission amplitude of electron teleportation, due to the “branch cuts” sweeping through the Majorana zero modes [4], provides a signature of their non-Abelian statistics.
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Figure 4
Measurement-based braiding. Schematic depiction of the initial state $| ψ_{i} 〉$ , with Majorana zero modes $χ_{1}$ and $χ_{2}$ initialized in the state $i χ_{1} χ_{2} = + 1$ . Performing the indicated sequence of measurements is equivalent to braiding Majorana fermions $γ_{2}$ and $γ_{3}$ up to a normalization factor.
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Figure 5
Measurement “decision tree”. Summary of the measurement protocol. If a measurement yields an undesirable outcome, the previous measurement step may be repeated (as indicated) to recover the state before the undesirable measurement was performed.
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Figure 6
Experimental realization. Protocol for teleportation-based braiding without braiding is illustrated in a nanowire-based Majorana platform. A nanowire hosts six Majorana zero modes at the interface between topological and trivial superconducting regions. $γ_{1}, \dots, γ_{4}$ are used as topological qubits and $χ_{1}, χ_{2}$ as an ancilla qubit. The green strip can be either a normal metal with a long phase-coherence length or a Majorana island in a definite parity state (a Majorana bus). We begin by initializing the ancilla qubit in (a) before performing measurements of the appropriate Majorana bilinears in the top nanowire. The coupling between the topological superconductor wire to the normal metal or Majorana bus through the metallic strips may be turned on and off, as indicated schematically by the “switches”. Fluxes may be applied through appropriate loops, as indicated in (b), (c) and (d) for readout of the appropriate topological qubit via conductance or persistent current measurements.
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