FINDING: Non-Abelian anyon fusion algebras exhibit self-similarity that mirrors the braid group structure, enabling fault-tolerant 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 $|i-j|>1$ . - Fusion algebra: $\phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c$ , where $N_{ab}^c$ are non-negative integers (fusion coefficients). For non-Abelian anyons, $\sum_c N_{ab}^c > 1$ for some $a,b$ . - Fibonacci anyon model: single nontrivial particle $\tau$ with fusion rule $\tau \times \tau = 1 + \tau$ , leading to quantum dimension $d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618$ . - Braiding matrices yield phases $e^{i\theta}$ with $\theta = 2\pi/5, 4\pi/5$ (related to golden angle $2\pi/\phi^2 \approx 137.5^\circ$ ). CONNECTION: - Fibonacci anyon quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Non-Abelian anyon fusion algebras exhibit self-similarity that mirrors the braid group structure, enabling fault-tolerant 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 $. - Fusion algebra: $ \phi_a \times \phi_b = \sum_c N_{ab}^c \phi_c $, where $ N_{ab}^c $ are non-negative integers (fusion coefficients). For non-Abelian anyons, $ \sum_c N_{ab}^c > 1 $ for some $ a,b $. - Fibonacci anyon model: single nontrivial particle $ \tau $ with fusion rule $ \tau \times \tau = 1 + \tau $, leading to quantum dimension $ d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 $. - Braiding matrices yield phases $ e^{i\theta} $ with $ \theta = 2\pi/5, 4\pi/5 $ (related to golden angle $ 2\pi/\phi^2 \approx 137.5^\circ $). CONNECTION: - Fibonacci anyon quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 20, 20260 citations

Toric Code: Topological Order, Anyon, Quantum Error Correction (QEC) Codes and Fault-Tolerant Quantum Computation

Kishan Mishra

Sep 6, 20260 citations

Fault-Tolerant Quantum Error Correction for non-Abelian Anyons

Guillaume Dauphinais, David Poulin

arXiv:1607.02159Communications in Mathematical PhysicsJul 10, 201739 citations

Quantum computing with anyons is fault tolerant

Anasuya Lyons, Benjamin J. Brown

In seminal work (arxiv:quant-ph/9707021) Alexei Kitaev proposed topological quantum computing (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:quant-ph/0001108, arXiv:0707.1889), whereby logic gates of a quantum computer are conducted by creating, braiding and fusing anyonic particles on a two-dimensional plane. Furthermore, he showed the proposal is inherently robust to local perturbations (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:1001.0344, arXiv:1001.4363) when anyons are created as quasiparticle excitations of a topologically ordered lattice model prepared at zero temperature. Over the decades following this proposal there have been considerable technological developments towards the construction of a fault-tolerant quantum computer. Rather than maintaining some target ground state at zero temperature, a modern approach is to actively correct the errors a target state experiences, where we use noisy quantum circuit elements to identify and subsequently correct for deviations from the ideal state. We present an error-correction scheme that enables us to carry out robust universal quantum computation by braiding anyons. We show that our scheme can be carried out on a suitably large device with an arbitrarily small failure rate assuming circuit elements are below some threshold level of local noise. The error-corrected scheme we have developed therefore enables us to carry out fault-tolerant topological quantum computation using modern quantum hardware that is now under development.In seminal work (arxiv:quant-ph/9707021) Alexei Kitaev proposed topological quantum computing (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:quant-ph/0001108, arXiv:0707.1889), whereby logic gates of a quantum computer are conducted by creating, braiding and fusing anyonic particles on a two-dimensional plane. Furthermore, he showed the proposal is inherently robust to local perturbations (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:1001.0344, arXiv:1001.4363) when anyons are created as quasiparticle excitations of a topologically ordered lattice model prepared at zero temperature. Over the decades following this proposal there have been considerable technological developments towards the construction of a fault-tolerant quantum computer. Rather than maintaining some target ground state at zero temperature, a modern approach is to actively correct the errors a target state experiences, where we use noisy quantum circuit elements to identify and subsequently correct for deviations from the ideal state. We present an error-correction scheme that enables us to carry out robust universal quantum computation by braiding anyons. We show that our scheme can be carried out on a suitably large device with an arbitrarily small failure rate assuming circuit elements are below some threshold level of local noise. The error-corrected scheme we have developed therefore enables us to carry out fault-tolerant topological quantum computation using modern quantum hardware that is now under development.

quant-pharXiv:2602.11258Feb 13, 20260 citations

Practicality of Fault-Tolerant Quantum Computation

Simon J. Devitt, Austin G. Fowler, Lloyd C. L. Hollenberg

Jan 1, 20050 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

. - Fusion algebra:

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

, where

N_{ab}^c

are non-negative integers (fusion coefficients). For non-Abelian anyons,

\sum_c N_{ab}^c > 1

for some

a,b

. - Fibonacci anyon model: single nontrivial particle

\tau

with fusion rule

\tau \times \tau = 1 + \tau

, leading to quantum dimension

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

. - Braiding matrices yield phases

e^{i\theta}

with

\theta = 2\pi/5, 4\pi/5

(related to golden angle

2\pi/\phi^2 \approx 137.5^\circ

). CONNECTION: - Fibonacci anyon quant Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com