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Europhysics Letters (EPL)

Quantum simulation of non-unitary operators generated by R -matrices in the Yang-Baxter equation

Chao Zheng, Jiamin Zhou, Jiatong Cheng, Guanru Feng

Published May 21, 2026• 0 citations
Publisher

Abstract

Abstract As a type of non-Abelian anyons, Yang-Lee anyons have topological quantum computing potential comparable to that of Fibonacci anyons, and their non-unitary braiding property is conducive to the construction of non-unitary quantum gates. Any braiding process of Yang-Lee anyons can be realized by combining two basic braiding matrices, which are generated by the R -matrix of the Yang-Baxter equation. However, one of the basic braiding matrices of the braiding process of Yang-Lee anyons is difficult to be simulated directly because of its non-unitary property. To address this challenge, we propose, using the linear combination of unitaries (LCU) framework, the first probabilistic simulation scheme for the three-dimensional non-unitary matrix associated with Yang-Lee anyon braiding. We design two feasible quantum circuits in a qubit-qutrit hybrid system and in a pure-qubit system. Both schemes achieve simulation through controlled operations and ancillary qubit measurements, and the success probability is jointly determined by the input state, the dimensions of the total Hilbert space and the normalization factor. This work presents an approach to simulating the three-dimensional non-unitary matrix associated with Yang-Lee anyon braiding, which can serve as a building block for non-unitary quantum information processing.

DOI
10.1209/0295-5075/ae7152

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As a type of non-Abelian anyons, Yang-Lee anyons have topological quantum computing potential comparable to that of Fibonacci anyons, and their non-unitary braiding property is conducive to the construction of non-unitary quantum gates. Any braiding process of Yang-Lee anyons can be realized by combining two basic braiding matrices, which are generated by the R-matrix of the Yang-Baxter equation. However, one of the basic braiding matrices of the braiding process of Yang-Lee anyons is difficult to be simulated directly because of its non-unitary property. To address this challenge, we propose, using the linear combination of unitaries (LCU) framework, the first probabilistic simulation scheme for the three-dimensional non-unitary matrix $\rho \left (\delta _{2}\right )$ associated with Yang-Lee anyon braiding. We design two feasible quantum circuits in a qubit-qutrit hybrid system and in a pure-qubit system. Both schemes achieve simulation through controlled operations and ancillary qubit measurements, and the success probability is jointly determined by the input state, the dimensions of the total Hilbert space and the normalization factor. This work presents an approach to simulating the three-dimensional non-unitary matrix associated with Yang-Lee anyon braiding, which can serve as a building block for non-unitary quantum information processing.

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Non-Abelian Anyons and Braid Group Representations for Topological Quantum Computation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | MATH: 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); non-Abelian anyons carry unitary braid matrices \(U(\sigma_i)\) with \(U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})\); Fibonacci anyons have fusion rule \(\tau \times \tau = 1 + \tau\), yielding quantum dimension \(\phi = (1+\sqrt{5})/2 \approx 1.618\) (golden ratio); topological protection arises from ground-state degeneracy on genus-\(g\) surfaces: \(\dim \mathcal{H} = 4^g\) for toric code, or \(2^g\) for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio \(\phi = 1.618\), and its inverse \(\phi^{-1} = 0.618\) appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Systematic Computation of Braid Generator Matrix in Topological Quantum Computing

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We provide a comprehensive systematic method for the numerical computation of elementary braid operations in topological quantum computation (TQC). This {procedure} is systematically applicable to all anyon models, including $SU(2)_k$. Braiding non-abelian anyons is the essence of TQC, offering a topologically protected implementation of quantum gates. However, obtaining elementary braid matrix representations starting from the fusion and rotation matrices of a specific anyon model is {theoretically guarenteed but no numerical method is given, especially for systems with numerous anyons and complex fusion patterns. Our proposed method addresses this challenge, first in the special case of sparse encoding, allowing for the inclusion of an arbitrary number of anyons per qudit, {and in the general case}. This is accomplished by introducing two methods, one is based on a novel braiding move we call knitting, {the second introduces more general algorithm which is optimal in number of required moves}. The method plays a key role in a broad topological quantum circuit simulator, enabling the examination and study of complex quantum circuits within the TQC framework. Importantly, it proves effective across various anyonic models, accommodating diverse fusion rules. We validate the method by simulating an approximated CNOT gate and present a first-of-a-kind GHZ state simulation on five qubits using three Fibonacci anyons per qubit.

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Non-Abelian Anyon Braiding for Topological Quantum Gates — E8 Intelligence Research

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FINDING: Anyon braiding for topological qubits relies on non-Abelian statistics in 2D systems, with quantum gates implemented via worldline braids in spacetime. MATH: Braid group B_n generators σ_i satisfy σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} (Yang-Baxter equation). Unitary representations yield quantum gates. Fusion rules: for Fibonacci anyons, τ × τ = 1 + τ, with quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618. Braiding matrices involve golden ratio: e.g., F-matrix entries are φ^{-1/2} and φ^{-3/2}. CONNECTION: Fibonacci anyon quantum dimension is exactly the golden ratio φ = 1.618. Braiding phases involve φ^{-1} ≈ 0.618 and φ^{-2} ≈ 0.382. The Yang-Baxter equation is a 2D solvability condition linked to root systems of Lie algebras (e.g., A_n, B_n, C_n, D_n). Anyon statistics arise from 2D topology, not 3D; this mirrors the exceptional role of 2D in crystallographic symmetry (17 wallpaper groups, 5 Bravais lattices in 2D). DEPTH: 8 — Directly ties topological quantum computation to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Abstract Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.

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Classical Simulation of Yang-Baxter Gates

Gorjan Alagic, Aniruddha Bapat, Stephen P. Jordan

A unitary operator that satisfies the constant Yang-Baxter equation immediately yields a unitary representation of the braid group B_n for every n >= 2. If we view such an operator as a quantum-computational gate, then topological braiding corresponds to a quantum circuit. A basic question is when such a representation affords universal quantum computation. In this work, we show how to classically simulate these circuits when the gate in question belongs to certain families of solutions to the Yang-Baxter equation. These include all of the qubit (i.e., d = 2) solutions, and some simple families that include solutions for arbitrary d >= 2. Our main tool is a probabilistic classical algorithm for efficient simulation of a more general class of quantum circuits. This algorithm may be of use outside the present setting.

arXiv:1407.1361•Leibniz international proceedings in informatics•Jan 1, 2014•4 citations
Abstract As a type of non-Abelian anyons, Yang-Lee anyons have topological quantum computing potential comparable to that of Fibonacci anyons, and their non-unitary braiding property is conducive to the construction of non-unitary quantum gates. Any braiding process of Yang-Lee anyons can be realized by combining two basic braiding matrices, which are generated by the R -matrix of the Yang-Baxter equation. However, one of the basic braiding matrices of the braiding process of Yang-Lee anyons is difficult to be simulated directly because of its non-unitary property. To address this challenge, we propose, using the linear combination of unitaries (LCU) framework, the first probabilistic simulation scheme for the three-dimensional non-unitary matrix associated with Yang-Lee anyon braiding. We design two feasible quantum circuits in a qubit-qutrit hybrid system and in a pure-qubit system. Both schemes achieve simulation through controlled operations and ancillary qubit measurements, and the success probability is jointly determined by the input state, the dimensions of the total Hilbert space and the normalization factor. This work presents an approach to simulating the three-dimensional non-unitary matrix associated with Yang-Lee anyon braiding, which can serve as a building block for non-unitary quantum information processing.
As a type of non-Abelian anyons, Yang-Lee anyons have topological quantum computing potential comparable to that of Fibonacci anyons, and their non-unitary braiding property is conducive to the construction of non-unitary quantum gates. Any braiding process of Yang-Lee anyons can be realized by combining two basic braiding matrices, which are generated by the R-matrix of the Yang-Baxter equation. However, one of the basic braiding matrices of the braiding process of Yang-Lee anyons is difficult to be simulated directly because of its non-unitary property. To address this challenge, we propose, using the linear combination of unitaries (LCU) framework, the first probabilistic simulation scheme for the three-dimensional non-unitary matrix ρ(δ2)\rho \left (\delta _{2}\right )ρ(δ2​) associated with Yang-Lee anyon braiding. We design two feasible quantum circuits in a qubit-qutrit hybrid system and in a pure-qubit system. Both schemes achieve simulation through controlled operations and ancillary qubit measurements, and the success probability is jointly determined by the input state, the dimensions of the total Hilbert space and the normalization factor. This work presents an approach to simulating the three-dimensional non-unitary matrix associated with Yang-Lee anyon braiding, which can serve as a building block for non-unitary quantum information processing.
FINDING: Anyon braiding provides non-Abelian statistics for topological quantum computation, with braid group representations forming the computational basis. | 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​ (Yang–Baxter equation); non-Abelian anyons carry unitary braid matrices U(σi)U(\sigma_i)U(σi​) with U(σi)U(σi+1)U(σi)=U(σi+1)U(σi)U(σi+1)U(\sigma_i)U(\sigma_{i+1})U(\sigma_i) = U(\sigma_{i+1})U(\sigma_i)U(\sigma_{i+1})U(σi​)U(σi+1​)U(σi​)=U(σi+1​)U(σi​)U(σi+1​); Fibonacci anyons have fusion rule τ×τ=1+τ\tau \times \tau = 1 + \tauτ×τ=1+τ, yielding quantum dimension ϕ=(1+5)/2≈1.618\phi = (1+\sqrt{5})/2 \approx 1.618ϕ=(1+5​)/2≈1.618 (golden ratio); topological protection arises from ground-state degeneracy on genus-ggg surfaces: dim⁡H=4g\dim \mathcal{H} = 4^gdimH=4g for toric code, or 2g2^g2g for Ising anyons. | CONNECTION: Fibonacci anyons' quantum dimension is *exactly* the golden ratio ϕ=1.618\phi = 1.618ϕ=1.618, and its inverse ϕ−1=0.618\phi^{-1} = 0.618ϕ−1=0.618 appears in braid fusion probabilities; the Yang–Baxter equation is a braided analogue of th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We provide a comprehensive systematic method for the numerical computation of elementary braid operations in topological quantum computation (TQC). This {procedure} is systematically applicable to all anyon models, including SU(2)kSU(2)_kSU(2)k​. Braiding non-abelian anyons is the essence of TQC, offering a topologically protected implementation of quantum gates. However, obtaining elementary braid matrix representations starting from the fusion and rotation matrices of a specific anyon model is {theoretically guarenteed but no numerical method is given, especially for systems with numerous anyons and complex fusion patterns. Our proposed method addresses this challenge, first in the special case of sparse encoding, allowing for the inclusion of an arbitrary number of anyons per qudit, {and in the general case}. This is accomplished by introducing two methods, one is based on a novel braiding move we call knitting, {the second introduces more general algorithm which is optimal in number of required moves}. The method plays a key role in a broad topological quantum circuit simulator, enabling the examination and study of complex quantum circuits within the TQC framework. Importantly, it proves effective across various anyonic models, accommodating diverse fusion rules. We validate the method by simulating an approximated CNOT gate and present a first-of-a-kind GHZ state simulation on five qubits using three Fibonacci anyons per qubit.
FINDING: Anyon braiding for topological qubits relies on non-Abelian statistics in 2D systems, with quantum gates implemented via worldline braids in spacetime. MATH: Braid group B_n generators σ_i satisfy σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1} (Yang-Baxter equation). Unitary representations yield quantum gates. Fusion rules: for Fibonacci anyons, τ × τ = 1 + τ, with quantum dimension d_τ = φ = (1+√5)/2 ≈ 1.618. Braiding matrices involve golden ratio: e.g., F-matrix entries are φ^{-1/2} and φ^{-3/2}. CONNECTION: Fibonacci anyon quantum dimension is exactly the golden ratio φ = 1.618. Braiding phases involve φ^{-1} ≈ 0.618 and φ^{-2} ≈ 0.382. The Yang-Baxter equation is a 2D solvability condition linked to root systems of Lie algebras (e.g., A_n, B_n, C_n, D_n). Anyon statistics arise from 2D topology, not 3D; this mirrors the exceptional role of 2D in crystallographic symmetry (17 wallpaper groups, 5 Bravais lattices in 2D). DEPTH: 8 — Directly ties topological quantum computation to Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Abstract Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.
A unitary operator that satisfies the constant Yang-Baxter equation immediately yields a unitary representation of the braid group B_n for every n >= 2. If we view such an operator as a quantum-computational gate, then topological braiding corresponds to a quantum circuit. A basic question is when such a representation affords universal quantum computation. In this work, we show how to classically simulate these circuits when the gate in question belongs to certain families of solutions to the Yang-Baxter equation. These include all of the qubit (i.e., d = 2) solutions, and some simple families that include solutions for arbitrary d >= 2. Our main tool is a probabilistic classical algorithm for efficient simulation of a more general class of quantum circuits. This algorithm may be of use outside the present setting.