Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Topological quantum computing and SU (2) braid group representations

Louis Hirsch Kauffman, Samuel J. Lomonaco

Published Mar 5, 2008• 2 citations

Publisher

Abstract

We study representations of the braid group to SU(2) and their relationships with topological quantum computation.

DOI: 10.1117/12.778068

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Topological quantum computing and SU (2) braid group representations

Louis Hirsch Kauffman, Samuel J. Lomonaco

Published Mar 5, 2008• 2 citations

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Abstract

We study representations of the braid group to SU(2) and their relationships with topological quantum computation.

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FINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling topological quantum computation via braiding operations. 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

. - Hecke algebra relations:

(\sigma_i - q^{1/2})(\sigma_i + q^{-1/2}) = 0

, with

q = e^{2\pi i/k}

for SU(2)_k anyons. - Fusion rules:

\phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c

, where

N_{ab}^c

are non-negative integers (fusion coefficients). - Fibonacci anyon fusion:

\tau \times \tau = 1 + \tau

, with quantum dimension

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

. CONNECTION: - Fibonacci anyon quantum dimension

d_\tau = \phi = 1.618

(golden ratio). - Braid matrix eigenvalues for Fibonacci anyons:

e^{\pm 3\pi i/5}

, related to golden ratio via \( 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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|\geq 2

. Non-Abelian anyons carry unitary representations

U(\sigma_i)

with

U(\sigma_i)^2 \neq I

(unlike fermions/bosons). Majorana zero modes obey

\gamma_i^\dagger = \gamma_i

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

, and a pair encodes a qubit via parity

i\gamma_1\gamma_2 = \pm 1

. Topological protection arises from the gap

\Delta E \sim e^{-L/\xi}

(exponential suppression with wire length

L

vs coherence length

\xi

). Fibonacci anyons (from SU(2)

_3

Chern-Simons theory) have quantum dimension

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

, giving dense braiding — universal quantum co Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com