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arXiv:1807.10414 · Mathematical Structures in Computer Science

Witness algebra and anyon braiding

Andreas Blass, Yuri Gurevich

Published Mar 1, 2020• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1017/s0960129520000055

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FINDING: Modular tensor categories (MTCs) provide the algebraic backbone for topological quantum computation, with S and T matrices encoding anyon braiding and fusion rules. | MATH: MTC data: fusion rules \(N^k_{ij}\), \(S\)-matrix (modular transformation \(\tau \to -1/\tau\)), \(T\)-matrix (diagonal, \(\theta_a = e^{2\pi i h_a}\) topological spins), satisfying \(S^2 = C\) (charge conjugation), \((ST)^3 = e^{2\pi i c/8} S^2\) (where \(c\) is the chiral central charge). The entries of \(S\) and \(T\) lie in a cyclotomic field \(\mathbb{Q}(\zeta_N)\), \(\zeta_N = e^{2\pi i/N}\). For the Fibonacci anyon MTC: \(S = \frac{1}{\sqrt{\phi+2}}\begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix}\), \(T = \text{diag}(1, e^{4\pi i/5})\), where \(\phi = (1+\sqrt{5})/2\). | CONNECTION: **Direct hit**: The Fibonacci MTC's \(S\)-matrix contains \(\phi = 1.618\) and \(\phi+2 = \phi^2+1 = 3.618\), whose inverse square root yields \(1/\sqrt{3.618} \approx 0.5257\). The golden ratio appears intrinsically. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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•Jul 31, 2013•4 citations
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.
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.
Quantum computing with anyons: an FFF-matrix and braid calculator
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 FFF-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.
This paper investigates quantum contextuality, a central nonclassical aspect of quantum mechanics, by employing the algebraic and topological structures of modular tensor categories. The analysis establishes that braid group representations constructed from modular categories, including the SU(2)k and Fibonacci anyon models, inherently produce state-dependent contextuality, as revealed by measurable violations of noncontextuality inequalities. The explicit construction of unitary representations on fusion spaces allows this paper to identify a direct structural correspondence between braiding operations and logical contextuality frameworks. The results offer a comprehensive topological framework to classify and quantify contextuality in low-dimensional quantum systems, thereby elucidating its role as a resource in topological quantum computation and advancing the interface between quantum algebra, topology, and quantum foundations.
Topological quantum computation offers a paradigm shift for fault-tolerant quantum computation. It proposes encoding qubits in the mutual states of quasiparticles (called anyons) of 2-dimensional topologically ordered matter. Unitary ribbon fusion categories describe the algebraic theory of anyons. Although the category of an anyonic theory specifies suitable computational spaces, qubits, and quantum gates, it does not precisely formalise the category of topological quantum computation (TQC). We describe the TQC category using the elements of categorical quantum mechanics (CQM). Given the qubit-based graphical calculus accompanying CQM, called the ZX-calculus, we present the details of the ZX-representation of Fibonacci and Ising anyons. Finally, we propose a bicategorical approach to convene anyonic theory and TQC in one (bi)category. In the second part of the thesis, we discuss how the bicategory of module categories formalises boundary types and their excitations. We show theorems that reduce the data to physically meaningful data. A 2-category capable of describing topological quantum computation with point-like and string-like excitations is speculated to be a fusion 2-category. We discuss the available definition in the literature. The paper particularly lacks a rigorous definition for additive 2-categories; in this thesis, we give a definition. We propose the idea of typing higher-ranked matrices in additive 2-categories. Lastly, we overview the complete definition of monoidal 2-categories and recover Kapranov and Voevodsky's definition from the algebraic definition of tricategory by Gurski. This definition and the detailed list of data are the complete ingredients physicists need for quantum computation with higher dimensional excitations.
FINDING: Modular tensor categories (MTCs) provide the algebraic backbone for topological quantum computation, with S and T matrices encoding anyon braiding and fusion rules. | MATH: MTC data: fusion rules NijkN^k_{ij}Nijk​, SSS-matrix (modular transformation τ→−1/τ\tau \to -1/\tauτ→−1/τ), TTT-matrix (diagonal, θa=e2πiha\theta_a = e^{2\pi i h_a}θa​=e2πiha​ topological spins), satisfying S2=CS^2 = CS2=C (charge conjugation), (ST)3=e2πic/8S2(ST)^3 = e^{2\pi i c/8} S^2(ST)3=e2πic/8S2 (where ccc is the chiral central charge). The entries of SSS and TTT lie in a cyclotomic field Q(ζN)\mathbb{Q}(\zeta_N)Q(ζN​), ζN=e2πi/N\zeta_N = e^{2\pi i/N}ζN​=e2πi/N. For the Fibonacci anyon MTC: S=1ϕ+2(1ϕϕ−1)S = \frac{1}{\sqrt{\phi+2}}\begin{pmatrix} 1 & \phi \\ \phi & -1 \end{pmatrix}S=ϕ+2​1​(1ϕ​ϕ−1​), T=diag(1,e4πi/5)T = \text{diag}(1, e^{4\pi i/5})T=diag(1,e4πi/5), where ϕ=(1+5)/2\phi = (1+\sqrt{5})/2ϕ=(1+5​)/2. | CONNECTION: **Direct hit**: The Fibonacci MTC's SSS-matrix contains ϕ=1.618\phi = 1.618ϕ=1.618 and ϕ+2=ϕ2+1=3.618\phi+2 = \phi^2+1 = 3.618ϕ+2=ϕ2+1=3.618, whose inverse square root yields 1/3.618≈0.52571/\sqrt{3.618} \approx 0.52571/3.618​≈0.5257. The golden ratio appears intrinsically. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Instead of constructing anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relations between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. More general situation is also discussed in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin-Wen types of lattice models and some chiral topological phases.