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Physics Letters A

The representation of braid group in mixed-boundary punctures model of ising anyons on the toric code

Yao Shen, Fulin Zhang

Published Nov 20, 2025• 0 citations
Publisher
DOI
10.1016/j.physleta.2025.131175

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Non-Abelian statistics with mixed-boundary punctures on the toric code

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Lattice defects on the toric code, like twists and punctures, can exhibit exotic behavior and have been suggested as viable building-block candidates for topological quantum computing. Here the authors build an encoding around a new defect, namely a mixed-boundary puncture. They show that this strategy allows for non-Abelian fusion and braiding properties reminiscent of Majorana exchange.

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Non-Abelian Anyons and Braid Group Representations for Topological Quantum Computation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group \(B_n\) generators \(\sigma_i\) satisfy \(\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}\) (Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices \(U(\sigma_i)\) with \(U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})\); Fibonacci anyons have fusion rule \(\tau \times \tau = 1 + \tau\), yielding quantum dimension \(\phi = (1+\sqrt{5})/2 \approx 1.618\) (golden ratio); topological protection arises from ground-state degeneracy on genus-\(g\) surfaces: \(\dim \mathcal{H} = 4^g\) for toric code, or \(2^g\) for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio \(\phi = 1.618\), and its inverse \(\phi^{-1} = 0.618\) appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Fault-tolerant mixed boundary punctures on the toric code

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Abstract Defects on the toric code, a well-known exactly solvable Abelian anyon model, can exhibit non-Abelian statistical properties, which can be classified into punctures and twists. Benhemou et al. [ Phys. Rev. A 105 042417 (2022)] introduced a mixed boundary puncture model that integrates the advantages of both punctures and twists. They proposed that non-Abelian properties could be realized in the symmetric subspace {|++〉, |−−〉}. This work demonstrates that the nontrivial antisymmetric subspace {|+−〉, |−+〉} also supports non-Abelian statistics. The mixed boundary puncture model is shown to be fault-tolerant in both subspaces, offering resistance to collective dephasing noise and collective rotation noise. In addition, we propose and validate a quantum information masking scheme within the three-partite mixed boundary puncture model.

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Braid Group and Quantum Group of Z(N)Model

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Chinese Science Bulletin•Jan 1, 1993•1 citation
FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: Braid group BnB_nBn​ generators σi\sigma_iσi​ satisfy σiσi+1σi=σi+1σiσi+1\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}σi​σi+1​σi​=σi+1​σi​σi+1​ (Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices U(σi)U(\sigma_i)U(σi​) with U(σi)U(σi+1)U(σi)=U(σi+1)U(σi)U(σi+1)U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})U(σi​)U(σi+1​)U(σi​)=U(σi+1​)U(σi​)U(σi+1​); Fibonacci anyons have fusion rule τ×τ=1+τ\tau \times \tau = 1 + \tauτ×τ=1+τ, yielding quantum dimension ϕ=(1+5)/2≈1.618\phi = (1+\sqrt{5})/2 \approx 1.618ϕ=(1+5​)/2≈1.618 (golden ratio); topological protection arises from ground-state degeneracy on genus-ggg surfaces: dim⁡H=4g\dim \mathcal{H} = 4^gdimH=4g for toric code, or 2g2^g2g for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio ϕ=1.618\phi = 1.618ϕ=1.618, and its inverse ϕ−1=0.618\phi^{-1} = 0.618ϕ−1=0.618 appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Abstract Defects on the toric code, a well-known exactly solvable Abelian anyon model, can exhibit non-Abelian statistical properties, which can be classified into punctures and twists. Benhemou et al. [ Phys. Rev. A 105 042417 (2022)] introduced a mixed boundary puncture model that integrates the advantages of both punctures and twists. They proposed that non-Abelian properties could be realized in the symmetric subspace {|++〉, |−−〉}. This work demonstrates that the nontrivial antisymmetric subspace {|+−〉, |−+〉} also supports non-Abelian statistics. The mixed boundary puncture model is shown to be fault-tolerant in both subspaces, offering resistance to collective dephasing noise and collective rotation noise. In addition, we propose and validate a quantum information masking scheme within the three-partite mixed boundary puncture model.