arXiv:2212.00831

Quantum computing with anyons: an $F$-matrix and braid calculator

Willie Aboumrad

Published Dec 1, 2022• 0 citations

Abstract

We introduce a pentagon equation solver, available as part of SageMath, and use it to construct braid group representations associated to certain anyon systems. We recall the category-theoretic framework for topological quantum computation to explain how these representations describe the sets of logical gates available to an anyonic quantum computer for information processing. In doing so, we avoid venturing deep into topological or conformal quantum field theory. Instead, we present anyons abstractly as sets of labels together with a collection of data satisfying a number of axioms, including the pentagon and hexagon equations, and explain how these data characterize ribbon fusion categories (RFCs). In the language of RFCs, our solver can produce $F$-matrices for anyon systems corresponding to multiplicity-free fusion rings arising in connection with the representation theory of quantum groups associated to simple Lie algebras with deformation parameter a root of unity.

DOI: 10.48550/arxiv.2212.00831

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arXiv:2212.00831

Quantum computing with anyons: an $F$ -matrix and braid calculatorQuantum computing with anyons: an $F$-matrix and braid calculator

Willie Aboumrad

Published Dec 1, 2022• 0 citations

arXiv HTML PDF

Abstract

F

-matrices for anyon systems corresponding to multiplicity-free fusion rings arising in connection with the representation theory of quantum groups associated to simple Lie algebras with deformation parameter a root of unity.

DOI: 10.48550/arxiv.2212.00831

Related papers

The ribbon category framework for topological quantum computing

Willie Aboumrad

This expository article supplies the mathematical background underpinning the braid representation calculator introduced in arXiv:2212.00831; those representations describe the sets of logic gates available to a topological quantum computer for processing encoded qubits. Assuming little background in category theory, we first recall the notion of a ribbon fusion category (RFC), collecting most of the necessary definitions. Then we discuss how certain RFCs arise from the representation theory of quantum groups. We explore the braiding in these categories in detail, since it is essential for the quantum computing application.This expository article supplies the mathematical background underpinning the braid representation calculator introduced in arXiv:2212.00831; those representations describe the sets of logic gates available to a topological quantum computer for processing encoded qubits. Assuming little background in category theory, we first recall the notion of a ribbon fusion category (RFC), collecting most of the necessary definitions. Then we discuss how certain RFCs arise from the representation theory of quantum groups. We explore the braiding in these categories in detail, since it is essential for the quantum computing application.

arXiv:2212.02921Dec 6, 20220 citations

Topological Quantum Gates with 5-Fold Symmetry from Non-Abelian Anyons — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Non-Abelian anyon braid group representations generate unitary gates whose eigenvalues can include roots of unity tied to 72° rotations (5-fold symmetry), linking topological quantum computation to pentagonal/icosahedral structures. | MATH: Braid group $B_n$ acts on fusion spaces via $R$ -matrix (braiding) and $F$ -matrix (fusion) satisfying Yang-Baxter: $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$ . For Fibonacci anyons (golden chain), the braid matrices have eigenvalues $e^{\pm 4\pi i/5}$ and $e^{\pm 2\pi i/5}$ — i.e., 72° and 144° rotations. The quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ appears in the fusion rule $\tau \times \tau = 1 + \tau$ . | CONNECTION: **Direct hit.** 72° = $2\pi/5$ — the rotation angle of a regular pentagon. The Fibonacci anyon braid matrices are precisely the $Y$ -matrices of the Temperley-Lieb algebra at $q = e^{i\pi/5}$ , whose eigenvalues are $2\cos(\pi/5) = \phi$ and $2\cos(2\pi/5) = 1/\phi = \phi-1 = 0.618$ . The gol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Non-Abelian anyon braid group representations generate unitary gates whose eigenvalues can include roots of unity tied to 72° rotations (5-fold symmetry), linking topological quantum computation to pentagonal/icosahedral structures. | MATH: Braid group $B_n$ acts on fusion spaces via $R$-matrix (braiding) and $F$-matrix (fusion) satisfying Yang-Baxter: $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$. For Fibonacci anyons (golden chain), the braid matrices have eigenvalues $e^{\pm 4\pi i/5}$ and $e^{\pm 2\pi i/5}$ — i.e., 72° and 144° rotations. The quantum dimension $\phi = (1+\sqrt{5})/2 \approx 1.618$ appears in the fusion rule $\tau \times \tau = 1 + \tau$. | CONNECTION: **Direct hit.** 72° = $2\pi/5$ — the rotation angle of a regular pentagon. The Fibonacci anyon braid matrices are precisely the $Y$-matrices of the Temperley-Lieb algebra at $q = e^{i\pi/5}$, whose eigenvalues are $2\cos(\pi/5) = \phi$ and $2\cos(2\pi/5) = 1/\phi = \phi-1 = 0.618$. The gol Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Self-Similar Anyon Fusion Algebras Enable Topological Quantum Gates — E8 Intelligence Research

Andrew Stewart Caldin

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Fatimah Rita Ahmadi, Aleks Kissinger

Abstract Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang–Baxter equation is directly connected to an important rule in the complete ZX-calculus known as the P-rule, which enables one to interchange the phase gates defined with respect to complementary bases. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as <?CDATA $\pi/2$ ?> π / 2 Z- and X-phase gates, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.Abstract Unitary fusion categories formalise the algebraic theory of topological quantum computation. These categories come naturally enriched in a subcategory of the category of Hilbert spaces, and by looking at this subcategory, one can identify a collection of generators for implementing quantum computation. We represent such generators for the Fibonacci and Ising models, namely the encoding of qubits and the associated braid group representations, with the ZX-calculus and show that in both cases, the Yang–Baxter equation is directly connected to an important rule in the complete ZX-calculus known as the P-rule, which enables one to interchange the phase gates defined with respect to complementary bases. In the Ising case, this reduces to a familiar rule relating two distinct Euler decompositions of the Hadamard gate as <?CDATA $\pi/2$?> π / 2 Z- and X-phase gates, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.

arXiv:2211.03855Journal of Physics A Mathematical and TheoreticalAug 11, 20231 citation

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Colleen Delaney, Zhenghan Wang

We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.

arXiv:1811.02143Nov 6, 20182 citations

Quantum computing with anyons: an

F

-matrix and braid calculator

F

B_n

acts on fusion spaces via

R

-matrix (braiding) and

F

-matrix (fusion) satisfying Yang-Baxter:

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

. For Fibonacci anyons (golden chain), the braid matrices have eigenvalues

e^{\pm 4\pi i/5}

and

e^{\pm 2\pi i/5}

— i.e., 72° and 144° rotations. The quantum dimension

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

appears in the fusion rule

\tau \times \tau = 1 + \tau

. | CONNECTION: **Direct hit.** 72° =

2\pi/5

— the rotation angle of a regular pentagon. The Fibonacci anyon braid matrices are precisely the

Y

-matrices of the Temperley-Lieb algebra at

q = e^{i\pi/5}

, whose eigenvalues are

2\cos(\pi/5) = \phi

and

2\cos(2\pi/5) = 1/\phi = \phi-1 = 0.618

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

a \times b = \sum_c N_{ab}^c c

, where

N_{ab}^c

are non-negative integers (fusion coefficients). 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). For Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, with quantum dimension

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

. Braid matrices yield

R

-matrix eigenvalues:

e^{\pm i\pi/5}

e^{\pm i3\pi/5}

. CONNECTION: Fibonacci anyon quantum dimension

d_\tau = \phi = 1.618

(golden ratio). Fusion matrix eigenvalues involve

\phi^{-1} = 0.618

. Braid group representations correspond to

SU(2)_3

Chern-Simons theory, linked to

E_8

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

\pi/2

?> π / 2 Z- and X-phase gates, whereas in the Fibonacci case, we give a previously unconsidered exact solution of the P-rule involving the Golden ratio. We demonstrate the utility of these representations by giving graphical derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons. We furthermore present a fully graphical procedure for simulating and simplifying braids with the ZX-representation of Fibonacci anyons.