# Search for non-Abelian Majorana braiding statistics in superconductors

### Submission summary

 As Contributors: Carlo Beenakker Arxiv Link: https://arxiv.org/abs/1907.06497v2 (pdf) Date accepted: 2020-06-11 Date submitted: 2020-05-25 02:00 Submitted by: Beenakker, Carlo Submitted to: SciPost Physics Lecture Notes Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### 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".

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. Lect. Notes 15 (2020)

### Submission & Refereeing History

#### Published as SciPost Phys. Lect. Notes 15 (2020)

Submission 1907.06497v2 on 25 May 2020

## Reports on this Submission

### Anonymous Report 2 on 2020-6-7 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1907.06497v2, delivered 2020-06-07, doi: 10.21468/SciPost.Report.1744

### Report

This is a very well written and timely report. It pays attention to the impoortant open question of braiding and fusing Majorana fermions in the laboratory.
I have only some minor comments that the author can optionally address. These comments that have aminly to do with the presentation of the lecture notes are given as anotations on a PDF.

### Attachment

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### Report 1 by Benjamin Huard on 2020-6-4 (Invited Report)

• Cite as: Benjamin Huard, Report on arXiv:1907.06497v2, delivered 2020-06-04, doi: 10.21468/SciPost.Report.1738

### Strengths

This is an excellent version of the lecture notes that correspond to the courses that Carlo Beenakker gave during the Les Houches Summer School of July 2019.

None

### Report

I carefully read the first version of the notes arXiv:1907.06497 (v1) and found them excellent. I proposed a series of minor modifications that Carlo Beenakker implemented fully.
The only scientific issue I found was in Eq.(28) (in the new version Eq. (4.5)). In a transmon, the anharmonicity of the transmon does not lead to the Jaynes-Cummings relation that is written there but to a renormalized expression. For a standard transmon, the shift does not read $g^2/\Delta$ but $2E_C g^2/\delta^2$. In the new version, the expression is mentioned to be given within the Jaynes-Cummings model, which is now correct.

I am therefore very satisfied with the new version of the lecture notes and recommend them to be accepted as is.

• validity: top
• significance: top
• originality: top
• clarity: top
• formatting: perfect
• grammar: perfect