Bulletin of the American Physical Society

Milestones toward Majorana-based quantum computing: Fusion rule detection and topological qubit validation

Ryan V. Mishmash, David Aasen, Michael Hell, Andrew P. Higginbotham, Jeroen Danon, Martin Leijnse, Thomas Sand Jespersen, Joshua Folk, Charles M. Marcus, Karsten Flensberg, Jason Alicea

Published Mar 14, 2016• 0 citations

OpenAlex

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Bulletin of the American Physical Society

Milestones toward Majorana-based quantum computing: Fusion rule detection and topological qubit validation

Ryan V. Mishmash, David Aasen, Michael Hell, Andrew P. Higginbotham, Jeroen Danon, Martin Leijnse, Thomas Sand Jespersen, Joshua Folk, Charles M. Marcus, Karsten Flensberg, Jason Alicea

Published Mar 14, 2016• 0 citations

OpenAlex

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Lecture notes in networks and systemsJan 1, 20250 citations

Dynamical Signatures of Ground‐State Degeneracy to Discriminate against Andreev Levels in a Majorana Fusion Experiment

Aurélien Grabsch, Y. Cheipesh, C. W. J. Beenakker

Abstract Detection of the fusion rule of Majora na zero‐modes is a near‐term milestone on the road to topological quantum computation. An obstacle is that the non‐deterministic fusion outcome of topological zero‐modes can be mimicked by the merging of non‐topological Andreev levels. To distinguish these two scenarios, the dynamical signatures of the ground‐state degeneracy that is the defining property of non‐Abelian anyons is searched for. By adiabatically traversing parameter space along two different pathways, one can identify ground‐state degeneracies from the breakdown of adiabaticity. It is shown that the approach can discriminate against accidental degeneracies of Andreev levels.Abstract Detection of the fusion rule of Majora na zero‐modes is a near‐term milestone on the road to topological quantum computation. An obstacle is that the non‐deterministic fusion outcome of topological zero‐modes can be mimicked by the merging of non‐topological Andreev levels. To distinguish these two scenarios, the dynamical signatures of the ground‐state degeneracy that is the defining property of non‐Abelian anyons is searched for. By adiabatically traversing parameter space along two different pathways, one can identify ground‐state degeneracies from the breakdown of adiabaticity. It is shown that the approach can discriminate against accidental degeneracies of Andreev levels.

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FINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics for fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming topological qubit realization. | MATH: Braid group $B_n$ generators $\sigma_i$ satisfy $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$ and $\sigma_i \sigma_j = \sigma_j \sigma_i$ for $|i-j|>1$ . Anyon fusion rules: $\phi \times \phi = 1 + \phi$ (Fibonacci anyon model). Quantum dimension $d_\phi = \phi = (1+\sqrt{5})/2 \approx 1.618$ . Topological qubit encoded in fusion space dimension $F_{abc}^{d}$ (F-matrix) with entries involving golden ratio. Majorana zero modes obey $\gamma_i^2 = 1$ , $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$ . | CONNECTION: Fibonacci anyon quantum dimension $d_\phi = 1.618$ directly links to golden ratio. Braiding matrices yield entries proportional to $\phi^{-1} \approx 0.618$ and $\phi^{-2} \approx 0.382$ . The non-Abelian statistics corresp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing uses anyons with non-Abelian braiding statistics for fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming topological qubit realization. | MATH: Braid group $ B_n $ generators $ \sigma_i $ satisfy $ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} $ and $ \sigma_i \sigma_j = \sigma_j \sigma_i $ for $ |i-j|>1 $. Anyon fusion rules: $ \phi \times \phi = 1 + \phi $ (Fibonacci anyon model). Quantum dimension $ d_\phi = \phi = (1+\sqrt{5})/2 \approx 1.618 $. Topological qubit encoded in fusion space dimension $ F_{abc}^{d} $ (F-matrix) with entries involving golden ratio. Majorana zero modes obey $ \gamma_i^2 = 1 $, $ \{\gamma_i, \gamma_j\} = 2\delta_{ij} $. | CONNECTION: Fibonacci anyon quantum dimension $ d_\phi = 1.618 $ directly links to golden ratio. Braiding matrices yield entries proportional to $ \phi^{-1} \approx 0.618 $ and $ \phi^{-2} \approx 0.382 $. The non-Abelian statistics corresp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Aug 15, 20260 citations

B_n

generators

\sigma_i

satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

and

\sigma_i \sigma_j = \sigma_j \sigma_i

for

|i-j|>1

. Anyon fusion rules:

\phi \times \phi = 1 + \phi

(Fibonacci anyon model). Quantum dimension

d_\phi = \phi = (1+\sqrt{5})/2 \approx 1.618

. Topological qubit encoded in fusion space dimension

F_{abc}^{d}

(F-matrix) with entries involving golden ratio. Majorana zero modes obey

\gamma_i^2 = 1

\{\gamma_i, \gamma_j\} = 2\delta_{ij}

. | CONNECTION: Fibonacci anyon quantum dimension

d_\phi = 1.618

directly links to golden ratio. Braiding matrices yield entries proportional to

\phi^{-1} \approx 0.618

and

\phi^{-2} \approx 0.382

. The non-Abelian statistics corresp Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com