Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2501.01745 · Physical Review A

Topological quantum compilation of metaplectic anyons based on the genetic optimized algorithms

Jiangwei Long, Jianxin Zhong, Lijun Meng

Published Jun 26, 2025• 4 citations
arXivHTMLPDF

Abstract

Topological quantum computing holding global anti-interference ability is realized by braiding some anyons, such as well-known Fibonacci anyons. Here, based on $SO(3)_2 $ theory we obtain a total of 6 anyon models utilizing \textit{F}-matrices, \textit{R}-symbols, and fusion rules of metaplectic anyon.We obtain the elementary braiding matrices (EBMs) by means of unconventional encoding. After braiding \textit{X} and $X^\prime$, we insert a pair of \textit{Z} anyons into them to ensure that the initial order of anyons remains unchanged. In this process only fusion is required, and measurement is not necessary. Three of them $\{V^{113}_3,V^{131}_3,V^{133}_1\}$ are studied in detail. We study systematically the compilation of these three models through EBMs obtained analytically. For one-qubit case, the classical \textit{H}- and \textit{T}-gate can be well constructed using the genetic algorithm enhanced Solovay-Kitaev algorithm (GA-enhanced SKA) by $\{V^{113}_3,V^{131}_3,V^{133}_1\}$. The obtained accuracy of the \textit{H}/\textit{T}-gate by $\{V^{113}_3,V^{133}_1\}$ is slightly inferior to the corresponding gates of the Fibonacci anyon model, but it also can meet the requirements of fault-tolerant quantum computing, $V^{131}_3$ giving the best performance of these four models. For the two-qubit case, we use the exhaustive method for short lengths and the GA for long lengths to obtain braidword for $\{V^{113}_3,V^{131}_3,V^{133}_1\}$ models. The resulting matrices can well approximate the local equivalence class of the CNOT-gate, while demonstrating a much smaller error than the Fibonacci model, especially for the $V^{113}_3$.The braiding processes of conventional encoding (using identical anyons) and unconventional encoding (using distinct anyons) are compared. Finally, we attempt to generalize the model to the \textit{N}-qubit case.

DOI
10.1103/js85-r2m7

Related papers

Genetic algorithm enhanced Solovay-Kitaev algorithm for quantum compiling of Fibonacci anyons

Jiangwei Long, Xuyang Huang, Jianxin Zhong +1 more

Abstract Quantum compiling, which aims to approximate target qubit gates by finding optimal sequences (braidwords) of basic braid operations, constitutes a fundamental challenge in quantum computing. We develop a genetic algorithm (GA)-enhanced Solovay–Kitaev algorithm (SKA) for approximating single-qubit gates using four elementary braiding matrices (EBMs) derived from Fibonacci anyons. The GA-enhanced SKA demonstrates robust performance, efficiently identifying optimal braidwords within exponentially large search spaces. Notably, the approximation precision achieved by our method surpasses that of Monte Carlo (MC)-enhanced SKA and becomes comparable to deep reinforcement learning (RL) approaches when braidword lengths exceed 25. Implementing 2- and 3-order approximations with the GA-enhanced SKA yields optimal braidword (initial braiding lengths l 0 = 50 and 30 respectively) achieving gate distances of 5.9 × 10 −7 - sufficient precision for most quantum computing applications. This work develops an optimized compilation framework for non-Abelian anyon gates, providing an essential methodology for enhancing future topological quantum computation architectures through gate optimization.

arXiv:2501.01746•Physica Scripta•Nov 1, 2025•1 citation

Universal Quantum Gate Compilation in $SU(2)_k$ Anyon Models via Multiple-Braiding

Jiangwei Long, Zihui Liu, Yizhi Li +1 more

We investigate the capability of multiple-braiding in $SU(2)_k$ anyon models to realize universal quantum computation. The multiple elementary braiding matrices (MEBMs) are derived from the $q$-deformed representation theory of $SU(2)$. Through a general analysis of the MEBMs, we find that multiple-braiding loses its universality only when the braiding multiplicity $m$ causes the MEBMs to collapse to a scalar (up to a global phase). Our numerical analysis of the MEBMs of $SU(2)_k$ anyon models with $k = 3, 5, 6, 7$, and $m \in [2, 9]$, shows that the values of $m$ at which the braid group representation fails to be dense agree exactly with the theoretical prediction. For those $m$ that support universality, single-qubit gates are compiled to high precision by a genetic algorithm-enhanced Solovay-Kitaev algorithm (GA-enhanced SKA), and a genetic algorithm (GA) search with progressively increasing braid length yields an approximately the local equivalence class $[CNOT]$. Notably, even-order braiding operations offer a physical advantage by reducing the number of non-Abelian anyons required in braiding-based topological quantum computation (TQC). Our analytical and numerical results together provide strong evidence for identifying which braiding multiplicities $m$ support universal quantum computation in $SU(2)_k$ anyon models.

quant-pharXiv:2602.15324•Feb 18, 2026•0 citations

High-Fidelity Universal Quantum Gate Compilation for Non-semisimple Ising Anyons via Genetic Algorithm-Optimized Solovay-Kitaev Decomposition

Jiangwei Long, Zihui Liu, Yizhi Li +2 more

Non-semisimple Ising anyons extend the fusion space of conventional Ising models and offer a new route toward universal topological quantum computation (TQC). However, how to construct a high-fidelity universal quantum gate set that meets fault-tolerant thresholds via braiding operations of non-semisimple Ising anyons remains unexplored. By employing a Genetic Algorithm-enhanced Solovay--Kitaev Algorithm (GA-enhanced SKA), we achieve high-fidelity approximations of standard single-qubit gates (Hadamard $H$-gate and phase $T$-gate) with a recursion level of just three, meeting the fidelity requirements for fault-tolerant quantum computation. Our numerical results demonstrate that for the critical parameter range $α\in [2.001,\, 2.022]$, a few braiding operations can approximate the local equivalence class $[\mathrm{CNOT}]$ with high precision. In contrast to prior theoretical works that established the universality of non-semisimple Ising anyons, this work provides the first complete, executable, and parameter-optimized compilation scheme for high-fidelity universal gates. It quantifies the relationship between the key physical parameter $α$ and gate fidelity, and it delivers direct numerical evidence for the universality of non-semisimple Ising anyons.

quant-pharXiv:2511.13194•Journal of Physics A Mathematical and Theoretical•Nov 18, 2025•0 citations

Systematic Computation of Braid Generator Matrix in Topological Quantum Computing

Abdellah Tounsi, Nacer Eddine Belaloui, Mohamed Messaoud Louamri +3 more

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.

arXiv:2307.01892•Jul 4, 2023•0 citations

Topological quantum compiling

Layla Hormozi, Georgios Zikos, N. E. Bonesteel +1 more

A method for compiling quantum algorithms into specific braiding patterns for non-Abelian quasiparticles described by the so-called Fibonacci anyon model is developed. The method is based on the observation that a universal set of quantum gates acting on qubits encoded using triplets of these quasiparticles can be built entirely out of three-stranded braids (three-braids). These three-braids can then be efficiently compiled and improved to any required accuracy using the Solovay-Kitaev algorithm.

arXiv:quant-ph/0610111•Physical Review B•Apr 11, 2007•96 citations

Topological Quantum Computation and Quantum Compilation

Joren Brunekreef

A central problem in realizing universal quantum computers is dealing with decoherence effects, which can destroy the vulnerable quantum states present in the internal registers. Several schemes have been developed that provide resistance to decoherence, leading to theories of fault-tolerant quantum computation. One such scheme is the theory of topological quantum computation, which encodes qubit states in robust topological properties of certain quantum systems. However, fault-tolerant theories typically only provide a finite set of primitive gates, from which we must then form approximations to arbitrary quantum gates. The Solovay-Kitaev algorithm describes a procedure for accomplishing this, and proves that the number of primitive gates required to approximate an arbitrary gate is only polylogarithmic in the desired degree of accuracy. In this thesis, we first outline the basic theory of quantum computation. Secondly, we describe the general problem of quantum compilation, list several possible strategies for relevant optimizations, and outline a proof of the Solovay-Kitaev theorem. Next, we give an introduction to the theory of topological quantum computation by anyons, with special focus on Fibonacci-anyons. Finally, we list some numerical results of an implementation of the Solovay-Kitaev algorithm when applied to a Fibonacci-anyon model and analyze the performance gains obtained through applications of the proposed optimizations.

•Aug 26, 2014•0 citations
Topological quantum computing holding global anti-interference ability is realized by braiding some anyons, such as well-known Fibonacci anyons. Here, based on SO(3)2SO(3)_2 SO(3)2​ theory we obtain a total of 6 anyon models utilizing \textit{F}-matrices, \textit{R}-symbols, and fusion rules of metaplectic anyon.We obtain the elementary braiding matrices (EBMs) by means of unconventional encoding. After braiding \textit{X} and X′X^\primeX′, we insert a pair of \textit{Z} anyons into them to ensure that the initial order of anyons remains unchanged. In this process only fusion is required, and measurement is not necessary. Three of them {V3113,V3131,V1133}\{V^{113}_3,V^{131}_3,V^{133}_1\}{V3113​,V3131​,V1133​} are studied in detail. We study systematically the compilation of these three models through EBMs obtained analytically. For one-qubit case, the classical \textit{H}- and \textit{T}-gate can be well constructed using the genetic algorithm enhanced Solovay-Kitaev algorithm (GA-enhanced SKA) by {V3113,V3131,V1133}\{V^{113}_3,V^{131}_3,V^{133}_1\}{V3113​,V3131​,V1133​}. The obtained accuracy of the \textit{H}/\textit{T}-gate by {V3113,V1133}\{V^{113}_3,V^{133}_1\}{V3113​,V1133​} is slightly inferior to the corresponding gates of the Fibonacci anyon model, but it also can meet the requirements of fault-tolerant quantum computing, V3131V^{131}_3V3131​ giving the best performance of these four models. For the two-qubit case, we use the exhaustive method for short lengths and the GA for long lengths to obtain braidword for {V3113,V3131,V1133}\{V^{113}_3,V^{131}_3,V^{133}_1\}{V3113​,V3131​,V1133​} models. The resulting matrices can well approximate the local equivalence class of the CNOT-gate, while demonstrating a much smaller error than the Fibonacci model, especially for the V3113V^{113}_3V3113​.The braiding processes of conventional encoding (using identical anyons) and unconventional encoding (using distinct anyons) are compared. Finally, we attempt to generalize the model to the \textit{N}-qubit case.
Abstract Quantum compiling, which aims to approximate target qubit gates by finding optimal sequences (braidwords) of basic braid operations, constitutes a fundamental challenge in quantum computing. We develop a genetic algorithm (GA)-enhanced Solovay–Kitaev algorithm (SKA) for approximating single-qubit gates using four elementary braiding matrices (EBMs) derived from Fibonacci anyons. The GA-enhanced SKA demonstrates robust performance, efficiently identifying optimal braidwords within exponentially large search spaces. Notably, the approximation precision achieved by our method surpasses that of Monte Carlo (MC)-enhanced SKA and becomes comparable to deep reinforcement learning (RL) approaches when braidword lengths exceed 25. Implementing 2- and 3-order approximations with the GA-enhanced SKA yields optimal braidword (initial braiding lengths l 0 = 50 and 30 respectively) achieving gate distances of 5.9 × 10 −7 - sufficient precision for most quantum computing applications. This work develops an optimized compilation framework for non-Abelian anyon gates, providing an essential methodology for enhancing future topological quantum computation architectures through gate optimization.
Universal Quantum Gate Compilation in SU(2)kSU(2)_kSU(2)k​ Anyon Models via Multiple-Braiding
We investigate the capability of multiple-braiding in SU(2)kSU(2)_kSU(2)k​ anyon models to realize universal quantum computation. The multiple elementary braiding matrices (MEBMs) are derived from the qqq-deformed representation theory of SU(2)SU(2)SU(2). Through a general analysis of the MEBMs, we find that multiple-braiding loses its universality only when the braiding multiplicity mmm causes the MEBMs to collapse to a scalar (up to a global phase). Our numerical analysis of the MEBMs of SU(2)kSU(2)_kSU(2)k​ anyon models with k=3,5,6,7k = 3, 5, 6, 7k=3,5,6,7, and m∈[2,9]m \in [2, 9]m∈[2,9], shows that the values of mmm at which the braid group representation fails to be dense agree exactly with the theoretical prediction. For those mmm that support universality, single-qubit gates are compiled to high precision by a genetic algorithm-enhanced Solovay-Kitaev algorithm (GA-enhanced SKA), and a genetic algorithm (GA) search with progressively increasing braid length yields an approximately the local equivalence class [CNOT][CNOT][CNOT]. Notably, even-order braiding operations offer a physical advantage by reducing the number of non-Abelian anyons required in braiding-based topological quantum computation (TQC). Our analytical and numerical results together provide strong evidence for identifying which braiding multiplicities mmm support universal quantum computation in SU(2)kSU(2)_kSU(2)k​ anyon models.
Non-semisimple Ising anyons extend the fusion space of conventional Ising models and offer a new route toward universal topological quantum computation (TQC). However, how to construct a high-fidelity universal quantum gate set that meets fault-tolerant thresholds via braiding operations of non-semisimple Ising anyons remains unexplored. By employing a Genetic Algorithm-enhanced Solovay--Kitaev Algorithm (GA-enhanced SKA), we achieve high-fidelity approximations of standard single-qubit gates (Hadamard HHH-gate and phase TTT-gate) with a recursion level of just three, meeting the fidelity requirements for fault-tolerant quantum computation. Our numerical results demonstrate that for the critical parameter range α∈[2.001, 2.022]α\in [2.001,\, 2.022]α∈[2.001,2.022], a few braiding operations can approximate the local equivalence class [CNOT][\mathrm{CNOT}][CNOT] with high precision. In contrast to prior theoretical works that established the universality of non-semisimple Ising anyons, this work provides the first complete, executable, and parameter-optimized compilation scheme for high-fidelity universal gates. It quantifies the relationship between the key physical parameter ααα and gate fidelity, and it delivers direct numerical evidence for the universality of non-semisimple Ising anyons.
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.
A method for compiling quantum algorithms into specific braiding patterns for non-Abelian quasiparticles described by the so-called Fibonacci anyon model is developed. The method is based on the observation that a universal set of quantum gates acting on qubits encoded using triplets of these quasiparticles can be built entirely out of three-stranded braids (three-braids). These three-braids can then be efficiently compiled and improved to any required accuracy using the Solovay-Kitaev algorithm.
A central problem in realizing universal quantum computers is dealing with decoherence effects, which can destroy the vulnerable quantum states present in the internal registers. Several schemes have been developed that provide resistance to decoherence, leading to theories of fault-tolerant quantum computation. One such scheme is the theory of topological quantum computation, which encodes qubit states in robust topological properties of certain quantum systems. However, fault-tolerant theories typically only provide a finite set of primitive gates, from which we must then form approximations to arbitrary quantum gates. The Solovay-Kitaev algorithm describes a procedure for accomplishing this, and proves that the number of primitive gates required to approximate an arbitrary gate is only polylogarithmic in the desired degree of accuracy. In this thesis, we first outline the basic theory of quantum computation. Secondly, we describe the general problem of quantum compilation, list several possible strategies for relevant optimizations, and outline a proof of the Solovay-Kitaev theorem. Next, we give an introduction to the theory of topological quantum computation by anyons, with special focus on Fibonacci-anyons. Finally, we list some numerical results of an implementation of the Solovay-Kitaev algorithm when applied to a Fibonacci-anyon model and analyze the performance gains obtained through applications of the proposed optimizations.