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arXiv:1401.0506 · Quantum Information Processing

On some projective unitary qutrit gates

Claire Levaillant

Published May 28, 2025• 6 citations
arXivHTMLPDF

Abstract

Abstract As part of a protocol, we braid in a certain way six anyons of topological charges 222211 in the Kauffman-Jones version of SU(2) Chern-Simons theory at level 4. The gate we obtain is a braid for the usual qutrit 2222 but with respect to a different basis than the usual basis. With respect to that basis, the Freedman group of J. Phys. A: Math. Theor. 47, 285203 (2014) is identical to the D-group D(18, 1, 1; 2, 1, 1) from the Blichfeldt classification. We provide a physical realization of each Blichfeldt generator of the SU(3) finite subgroup D(18, 1, 1; 2, 1, 1). The link of this group to elementary particle physics was already well-known by the physicists. This paper uncovers its realization in quantum physics. Inspired by these new techniques for the qutrit, we are able to make new ancillas, namely $$\frac{1}{\sqrt{2}}(|1>\,+|3>)$$ 1 2 ( | 1 > + | 3 > ) and $$\frac{1}{\sqrt{2}}(|1>\,-|3>)$$ 1 2 ( | 1 > - | 3 > ) , for the qubit 1221.

DOI
10.1007/s11128-025-04768-9

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Abstract As part of a protocol, we braid in a certain way six anyons of topological charges 222211 in the Kauffman-Jones version of SU(2) Chern-Simons theory at level 4. The gate we obtain is a braid for the usual qutrit 2222 but with respect to a different basis than the usual basis. With respect to that basis, the Freedman group of J. Phys. A: Math. Theor. 47, 285203 (2014) is identical to the D-group D(18, 1, 1; 2, 1, 1) from the Blichfeldt classification. We provide a physical realization of each Blichfeldt generator of the SU(3) finite subgroup D(18, 1, 1; 2, 1, 1). The link of this group to elementary particle physics was already well-known by the physicists. This paper uncovers its realization in quantum physics. Inspired by these new techniques for the qutrit, we are able to make new ancillas, namely 12(∣1> +∣3>)\frac{1}{\sqrt{2}}(|1>\,+|3>)2​1​(∣1>+∣3>) 1 2 ( | 1 > + | 3 > ) and 12(∣1> −∣3>)\frac{1}{\sqrt{2}}(|1>\,-|3>)2​1​(∣1>−∣3>) 1 2 ( | 1 > - | 3 > ) , for the qubit 1221.
We present a way to physically realize a circulant 2-qubit entangling gate in the Kauffman–Jones version of SU(2) Chern–Simons theory at level 4. Our approach uses qubit and qutrit ancillas, braids, fusions and interferometric measurements. Our qubit is formed by four anyons of topological charges 1221. Among other 2-qubit entangling gates which we generate in the present work, we produce in particular the circulant gate [Formula: see text] where J denotes the permutation matrix associated with the cycle [Formula: see text] and I denotes the identity matrix.
Abstract Quantum gates built out of braid group elements form the building blocks of topological quantum computation. They have been extensively studied in SU(2)k quantum group theories, a rich source of examples of non-Abelian anyons such as the Ising (k = 2), Fibonacci (k = 3) and Jones-Kauffman (k = 4) anyons. We show that the fusion spaces of these anyonic systems can be precisely mapped to the product state zero modes of certain Nicolai-like supersymmetric spin chains. As a result, we can realize the braid group in terms of the product state zero modes of these supersymmetric systems. These operators kill all the other states in the Hilbert space, thus preventing the occurrence of errors while processing information, making them suitable for quantum computing.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly , from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
The model of a topological quantum computer is a promising one due to its natural resistance to noise and other errors. Operations in such a computer are implemented by braiding the trajectories of anyons. While it is easy to understand how to build one-qubit operations, two-qubit operations are more difficult. In arXiv:2412.20931 we suggested an approach to build such operations for a topological quantum computer based on SU(2) Chern-Simons theory with arbitrary level using cabling of knots. In this paper we discuss how this approach should be generalized to the SU(N) case, what the differences are, and which new problems arise.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly from separable basis states important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
FINDING: Topological quantum computing leverages non-Abelian anyons and braid group representations for inherently fault-tolerant quantum gates, with Microsoft's Majorana 1 chip as a physical milestone. | MATH: Braid group BnB_nBn​ generators σi\sigma_iσi​ satisfy σiσi+1σi=σi+1σiσi+1\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}σi​σi+1​σi​=σi+1​σi​σi+1​ and σiσj=σjσi\sigma_i\sigma_j = \sigma_j\sigma_iσi​σj​=σj​σi​ for ∣i−j∣≥2|i-j|\geq 2∣i−j∣≥2. Non-Abelian anyons carry unitary representations U(σi)U(\sigma_i)U(σi​) with U(σi)2≠IU(\sigma_i)^2 \neq IU(σi​)2=I (unlike fermions/bosons). Majorana zero modes obey γi†=γi\gamma_i^\dagger = \gamma_iγi†​=γi​, {γi,γj}=2δij\{\gamma_i,\gamma_j\} = 2\delta_{ij}{γi​,γj​}=2δij​, and a pair encodes a qubit via parity iγ1γ2=±1i\gamma_1\gamma_2 = \pm 1iγ1​γ2​=±1. Topological protection arises from the gap ΔE∼e−L/ξ\Delta E \sim e^{-L/\xi}ΔE∼e−L/ξ (exponential suppression with wire length LLL vs coherence length ξ\xiξ). Fibonacci anyons (from SU(2)3_33​ Chern-Simons theory) have quantum dimension ϕ=(1+5)/2≈1.618\phi = (1+\sqrt{5})/2 \approx 1.618ϕ=(1+5​)/2≈1.618, giving dense braiding — universal quantum co Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com