Abstract Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.Abstract Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.

arXiv:1802.06176Quantum Science and TechnologyJun 6, 2018135 citations

Classical simulation of a topological quantum computer

B.O. Field, Tapio Simula

Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots.Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots.

Feb 17, 20181 citation

Anyonic Quantum Computation

Ismail Uysal

Jan 3, 20120 citations

Fibonacci Anyons Enable Universal Quantum Computation via Hexagonal Symmetry — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Fibonacci anyons realize non-Abelian braiding statistics with quantum dimension φ, enabling universal topological quantum computation via the A₂ root system's hexagonal symmetry. MATH: - Quantum dimension of Fibonacci anyon: $d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618$ - Fusion rule: $\tau \times \tau = 1 + \tau$ (where τ is the Fibonacci anyon) - Braiding matrices satisfy the Yang–Baxter equation and produce the Fibonacci representation of the braid group $B_n$ - A₂ root system: 6 roots at 60° angles, hexagonal lattice, Coxeter number 3, Weyl group order 6 CONNECTION: - The quantum dimension φ is exactly the golden ratio, linking anyon statistics to the Fibonacci sequence and pentagonal/quasicrystalline symmetry. - The A₂ root system's hexagonal lattice (crystallographic symmetry) underlies the fusion and braiding structure of Fibonacci anyons; the anyon's fusion algebra corresponds to the representation theory of the quantum group $ U_q(sl_2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Fibonacci anyons realize non-Abelian braiding statistics with quantum dimension φ, enabling universal topological quantum computation via the A₂ root system's hexagonal symmetry. MATH: - Quantum dimension of Fibonacci anyon: \( d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618 $ - Fusion rule: $ \tau \times \tau = 1 + \tau $ (where τ is the Fibonacci anyon) - Braiding matrices satisfy the Yang–Baxter equation and produce the Fibonacci representation of the braid group $ B_n $ - A₂ root system: 6 roots at 60° angles, hexagonal lattice, Coxeter number 3, Weyl group order 6 CONNECTION: - The quantum dimension φ is exactly the golden ratio, linking anyon statistics to the Fibonacci sequence and pentagonal/quasicrystalline symmetry. - The A₂ root system's hexagonal lattice (crystallographic symmetry) underlies the fusion and braiding structure of Fibonacci anyons; the anyon's fusion algebra corresponds to the representation theory of the quantum group \( U_q(sl_2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 25, 20260 citations

Fault-tolerant quantum computation by anyons

Alexei Kitaev

arXiv:quant-ph/9707021Annals of PhysicsJan 1, 20037,215 citations

d = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618

- Fusion rule:

\tau \times \tau = 1 + \tau

(where τ is the Fibonacci anyon) - Braiding matrices satisfy the Yang–Baxter equation and produce the Fibonacci representation of the braid group

B_n

- A₂ root system: 6 roots at 60° angles, hexagonal lattice, Coxeter number 3, Weyl group order 6 CONNECTION: - The quantum dimension φ is exactly the golden ratio, linking anyon statistics to the Fibonacci sequence and pentagonal/quasicrystalline symmetry. - The A₂ root system's hexagonal lattice (crystallographic symmetry) underlies the fusion and braiding structure of Fibonacci anyons; the anyon's fusion algebra corresponds to the representation theory of the quantum group \( U_q(sl_2) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com