arXiv:2202.08207 · Physical review. B./Physical review. B

Compatibility of braiding and fusion on wire networks

Aaron Conlon, Johannes K. Slingerland

Published Jul 27, 2023• 2 citations

Abstract

Exchanging particles on graphs, or more concretely on networks of quantum wires, has been proposed as a means to perform fault-tolerant quantum computation. This was inspired by braiding of anyons in planar systems. However, exchanges on a graph are not governed by the usual braid group but instead by a graph braid group. By imposing compatibility of graph braiding with fusion of topological charges, we obtain generalized hexagon equations. We find the usual planar anyon solutions but also more general braid actions. We illustrate this with Abelian, Fibonacci, and Ising fusion rules.

DOI: 10.1103/physrevb.108.035150

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arXiv:2202.08207 · Physical review. B./Physical review. B

Compatibility of braiding and fusion on wire networks

Aaron Conlon, Johannes K. Slingerland

Published Jul 27, 2023• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevb.108.035150

Related papers

Extending the planar theory of anyons to quantum wire networks

Tomasz Maciążek, Aaron Conlon, Gert Vercleyen +1 more

The braiding of the worldlines of particles restricted to move on a network (graph) is governed by the graph braid group, which can be strikingly different from the standard braid group known from two-dimensional physics. It has been recently shown that imposing the compatibility of graph braiding with anyon fusion for anyons exchanging at a single wire junction leads to new types of anyon models with the braiding exchange operators stemming from solutions of certain generalised hexagon equations. In this work, we establish these graph-braided anyon fusion models for general wire networks. We show that the character of braiding strongly depends on the graph-theoretic connectivity of the given network. In particular, we prove that triconnected networks yield the same braiding exchange operators as the planar anyon models. In contrast, modular biconnected networks support independent braiding exchange operators in different modules. Consequently, such modular networks may lead to more efficient topological quantum computer circuits. Finally, we conjecture that the graph-braided anyon fusion models will possess the (generalised) coherence property where certain polygon equations determine the braiding exchange operators for an arbitrary number of anyons. We also extensively study solutions to these polygon equations for chosen low-rank multiplicity-free fusion rings, including the Ising theory, quantum double of \mathbb{Z}_{2} ℤ 2 , and Tambara-Yamagami models. We find numerous solutions that do not appear in the planar theory of anyons.The braiding of the worldlines of particles restricted to move on a network (graph) is governed by the graph braid group, which can be strikingly different from the standard braid group known from two-dimensional physics. It has been recently shown that imposing the compatibility of graph braiding with anyon fusion for anyons exchanging at a single wire junction leads to new types of anyon models with the braiding exchange operators stemming from solutions of certain generalised hexagon equations. In this work, we establish these graph-braided anyon fusion models for general wire networks. We show that the character of braiding strongly depends on the graph-theoretic connectivity of the given network. In particular, we prove that triconnected networks yield the same braiding exchange operators as the planar anyon models. In contrast, modular biconnected networks support independent braiding exchange operators in different modules. Consequently, such modular networks may lead to more efficient topological quantum computer circuits. Finally, we conjecture that the graph-braided anyon fusion models will possess the (generalised) coherence property where certain polygon equations determine the braiding exchange operators for an arbitrary number of anyons. We also extensively study solutions to these polygon equations for chosen low-rank multiplicity-free fusion rings, including the Ising theory, quantum double of \mathbb{Z}_{2} ℤ 2 , and Tambara-Yamagami models. We find numerous solutions that do not appear in the planar theory of anyons.

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Tomasz Maciążek, Aaron Conlon, Gert Vercleyen +1 more

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Non-Abelian Anyon Fusion Mirrors Braid Groups for Fault-Tolerant Quantum Computing — E8 Intelligence Research

Andrew Stewart Caldin

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

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Abstract Indistinguishability of particles is a fundamental principle of quantum mechanics 1 . For all elementary and quasiparticles observed to date—including fermions, bosons and Abelian anyons—this principle guarantees that the braiding of identical particles leaves the system unchanged 2,3 . However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions 4–8 . Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well-developed mathematical description of non-Abelian anyons and numerous theoretical proposals 9–22 , the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. Whereas efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasiparticles, superconducting quantum processors allow for directly manipulating the many-body wavefunction by means of unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons 9,10 , we implement a generalized stabilizer code and unitary protocol 23 to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of using the anyons for quantum computation and use braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about non-Abelian braiding and, through the future inclusion of error correction to achieve topological protection, could open a path towards fault-tolerant quantum computing.Abstract Indistinguishability of particles is a fundamental principle of quantum mechanics 1 . For all elementary and quasiparticles observed to date—including fermions, bosons and Abelian anyons—this principle guarantees that the braiding of identical particles leaves the system unchanged 2,3 . However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions 4–8 . Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well-developed mathematical description of non-Abelian anyons and numerous theoretical proposals 9–22 , the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. Whereas efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasiparticles, superconducting quantum processors allow for directly manipulating the many-body wavefunction by means of unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons 9,10 , we implement a generalized stabilizer code and unitary protocol 23 to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of using the anyons for quantum computation and use braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about non-Abelian braiding and, through the future inclusion of error correction to achieve topological protection, could open a path towards fault-tolerant quantum computing.

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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.comFINDING: 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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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

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

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

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