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arXiv:1502.00669

On quantum computation, anyons, and categories

Andreas R. Blass, Yuri G. Gurevich

Published Feb 2, 2015• 3 citations
arXivHTMLPDF

Abstract

We explain the use of category theory in describing certain sorts of anyons. Yoneda's lemma leads to a simplification of that description. For the particular case of Fibonacci anyons, we also exhibit some calculations that seem to be known to the experts but not explicit in the literature.
DOI
10.48550/arxiv.1502.00669

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Title: Missing Direct Computation of U_q(sl_5) Level 3 Fibonacci Anyon Fusion Dimension — E8 Intelligence Research

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FINDING: The search results are largely tangential — no direct computation of the Fibonacci anyon fusion space dimension for U_q(sl_5) at level 3 is given. The closest mathematical anchor is the abstract anyon condensation paper (arXiv:1307.8244), which treats modular tensor categories abstractly, and the lecture titles on fusion categories of quantum groups, but no explicit dimension or quantum group data is extracted. The Fibonacci/golden ratio video is generic, not tied to anyons. MATH: No explicit equations or constants for U_q(sl_5) level 3 appear in the provided snippets. However, from standard theory (not in the search results but necessary for precision): For U_q(sl_N) at level k, the fusion category is a modular tensor category with simple objects labelled by Young diagrams in an (N-1)-rowed box of height k. For N=5, k=3, the number of simple objects is the number of partitions fitting in a 4×3 box = C(7,3) = 35. The Fibonacci anyon (τ) arises specifically in U_q(sl_2) at lev Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Oct 25, 2026•0 citations
To the working physicist, anyon theory is meant to describe certain quasi-particle excitations occurring in two dimensional topologically ordered systems. A typical calculation using this theory will involve operations such as ⊗\otimes⊗ to combine anyons, FdabcF^{abc}_{d}Fdabc​ to re-associate such combinations, and RcabR^{ab}_cRcab​ to commute or braid these anyons. Although there is a powerful string-diagram notation that greatly assists these manipulations, we still appear to be operating on particles arranged on a one-dimensional line, algebraically ordered from left to right. The obvious question is, where is the other dimension? The topological framework for considering these anyons as truly living in a two dimensional space is known as a modular functor, or topological quantum field theory. In this work we show how the apparently one-dimensional algebraic anyon theory is secretly the theory of anyons living in a fully two-dimensional system. The mathematical literature covering this secret is vast, and we try to distill this down into something more manageable.
Abstract Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.
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 π/2\pi/2π/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.
FINDING: The search results are largely tangential — no direct computation of the Fibonacci anyon fusion space dimension for U_q(sl_5) at level 3 is given. The closest mathematical anchor is the abstract anyon condensation paper (arXiv:1307.8244), which treats modular tensor categories abstractly, and the lecture titles on fusion categories of quantum groups, but no explicit dimension or quantum group data is extracted. The Fibonacci/golden ratio video is generic, not tied to anyons. MATH: No explicit equations or constants for U_q(sl_5) level 3 appear in the provided snippets. However, from standard theory (not in the search results but necessary for precision): For U_q(sl_N) at level k, the fusion category is a modular tensor category with simple objects labelled by Young diagrams in an (N-1)-rowed box of height k. For N=5, k=3, the number of simple objects is the number of partitions fitting in a 4×3 box = C(7,3) = 35. The Fibonacci anyon (τ) arises specifically in U_q(sl_2) at lev Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com