Majorana particles with noise: problems with braiding anyons

I will begin with a general discussion of anyons, of which Majoranas are an example. I continue with a discussion of why particle statistics in 2D can be neither bosonic nor fermionic, nor even abelian, and what it means for the charge to be fractional. A simple model of Kitaev, of electrons hopping on a branched 1D lattice with strong spin-orbit coupling and superconducting pairing, is sufficient to describe Majoranas and the gates that can be done by moving them around each other (braiding). I discuss a model calculation of the decohence of these Majorana qubits during braiding in a trijunction, due to thermally generated quasiparticles (bosonic environment). The limitations to coherence are significant; topological protection doesn't really work in this scenario. (work with Dr. Fabio Pedrocchi)I will begin with a general discussion of anyons, of which Majoranas are an example. I continue with a discussion of why particle statistics in 2D can be neither bosonic nor fermionic, nor even abelian, and what it means for the charge to be fractional. A simple model of Kitaev, of electrons hopping on a branched 1D lattice with strong spin-orbit coupling and superconducting pairing, is sufficient to describe Majoranas and the gates that can be done by moving them around each other (braiding). I discuss a model calculation of the decohence of these Majorana qubits during braiding in a trijunction, due to thermally generated quasiparticles (bosonic environment). The limitations to coherence are significant; topological protection doesn't really work in this scenario. (work with Dr. Fabio Pedrocchi)

Jan 1, 20170 citations

Reading the quantum state change of Majorana braiding

Chun-Xiao Liu, Ching‐Kai Chiu

Bulletin of the American Physical SocietyMar 4, 20190 citations

Tensor Network Algorithms for Braiding Anyons

Babatunde M. Ayeni, Sukhwinder Singh, Robert N. C. Pfeifer +1 more

Bulletin of the American Physical SocietyMar 17, 20160 citations

Topological Qubits via Non-Abelian Anyons and Braid Groups — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) and braid group representations to encode fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming hardware 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 $|i-j|>1$ . Anyonic exchange yields unitary matrices $U(\sigma_i)$ with non-commuting phases $e^{i\theta}$ , where $\theta$ depends on topological charge (e.g., Ising anyons: $\theta = \pi/8$ for $\sigma$ particles). Fibonacci anyons: dimension $d = \phi = (1+\sqrt{5})/2 \approx 1.618$ , with fusion rules $\tau \times \tau = 1 + \tau$ . Majorana zero modes: $\gamma_i^\dagger = \gamma_i$ , $\{\gamma_i, \gamma_j\} = 2\delta_{ij}$ , parity $P = i^k \prod \gamma_i$ . | CONNECTION: Fibonacci anyon dimension $d = \phi = 1.618$ — direct golden ratio e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological quantum computing leverages non-Abelian anyons (Majorana zero modes) and braid group representations to encode fault-tolerant qubits, with Microsoft's Majorana 1 chip claiming hardware 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 $. Anyonic exchange yields unitary matrices $ U(\sigma_i) $ with non-commuting phases $ e^{i\theta} $, where $ \theta $ depends on topological charge (e.g., Ising anyons: $ \theta = \pi/8 $ for $ \sigma $ particles). Fibonacci anyons: dimension $ d = \phi = (1+\sqrt{5})/2 \approx 1.618 $, with fusion rules $ \tau \times \tau = 1 + \tau $. Majorana zero modes: $ \gamma_i^\dagger = \gamma_i $, $ \{\gamma_i, \gamma_j\} = 2\delta_{ij} $, parity $ P = i^k \prod \gamma_i $. | CONNECTION: Fibonacci anyon dimension $ d = \phi = 1.618 $ — direct golden ratio e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Sep 8, 20260 citations

Majorana 1: Ising Anyons and the Unconfirmed Braid — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Microsoft's Majorana 1 chip claims topological qubit hardware based on Majorana zero modes, but the mathematical core is the non-Abelian braiding statistics of Ising anyons — not yet experimentally confirmed at scale. | MATH: Topological qubits encode information in the degenerate ground-state manifold of a 2D p-wave superconductor. The braid group $B_n$ acts on this space via unitary representations. For Ising anyons, the fusion rule is $\sigma \times \sigma = 1 + \psi$ , with quantum dimension $d_\sigma = \sqrt{2}$ . The braid matrices are $R$ -matrices satisfying the Yang–Baxter equation $R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$ . The topological charge is $e^{i\pi/8}$ for the $\sigma$ particle. Key constant: the Fibonacci anyon (if used instead) has golden-ratio quantum dimension $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . | CONNECTION: The Ising anyon's $\sqrt{2}$ quantum dimension links to the square lattice and $D_4$ root system ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Microsoft's Majorana 1 chip claims topological qubit hardware based on Majorana zero modes, but the mathematical core is the non-Abelian braiding statistics of Ising anyons — not yet experimentally confirmed at scale. | MATH: Topological qubits encode information in the degenerate ground-state manifold of a 2D p-wave superconductor. The braid group $ B_n $ acts on this space via unitary representations. For Ising anyons, the fusion rule is $ \sigma \times \sigma = 1 + \psi $, with quantum dimension $ d_\sigma = \sqrt{2} $. The braid matrices are $ R $-matrices satisfying the Yang–Baxter equation $ R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12} $. The topological charge is $ e^{i\pi/8} $ for the $ \sigma $ particle. Key constant: the Fibonacci anyon (if used instead) has golden-ratio quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. | CONNECTION: The Ising anyon's $ \sqrt{2} $ quantum dimension links to the square lattice and $ D_4 $ root system ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 13, 20260 citations

Braiding of majorana fermions and topological quantum computation

Louis Hirsch Kauffman

Journal of Material Science & EngineeringNov 16, 20160 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

. Anyonic exchange yields unitary matrices

U(\sigma_i)

with non-commuting phases

e^{i\theta}

, where

\theta

depends on topological charge (e.g., Ising anyons:

\theta = \pi/8

for

\sigma

particles). Fibonacci anyons: dimension

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

, with fusion rules

\tau \times \tau = 1 + \tau

. Majorana zero modes:

\gamma_i^\dagger = \gamma_i

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

, parity

P = i^k \prod \gamma_i

. | CONNECTION: Fibonacci anyon dimension

d = \phi = 1.618

— direct golden ratio e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

B_n

acts on this space via unitary representations. For Ising anyons, the fusion rule is

\sigma \times \sigma = 1 + \psi

, with quantum dimension

d_\sigma = \sqrt{2}

. The braid matrices are

R

-matrices satisfying the Yang–Baxter equation

R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}

. The topological charge is

e^{i\pi/8}

for the

\sigma

particle. Key constant: the Fibonacci anyon (if used instead) has golden-ratio quantum dimension

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

. | CONNECTION: The Ising anyon's

\sqrt{2}

quantum dimension links to the square lattice and

D_4

root system ( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com