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:2509.02843 · Physical Review A

Robust universal braiding with nonsemisimple Ising anyons

Filippo Iulianelli, Sung Kim, Joshua Sussan, Aaron D. Lauda

Published Mar 27, 2026• 0 citations
arXivHTMLPDF

Abstract

Non-semisimple extensions of the Ising anyon model developed in our previous work enable universal topological quantum computation via braiding alone, overcoming the Clifford-only limitation of semisimple theories. The non-semisimple theory provides new anyon types indexed by a real parameter $α$, the neglecton. Braiding acts unitarily with respect to an indefinite Hermitian form, while the computational subspace sits in a positive-definite sector. We demonstrate that this universality is robust, persisting over an open interval of the neglecton parameter $α$ where the computational subspace remains positive-definite. We identify special values of $α$ where the physical subspace decouples exactly from negative-norm components, ensuring fully unitary evolution and suppressed leakage. We further present an alternative encoding supporting exact single-qubit Clifford gates alongside a non-Clifford phase gate. We show that high-precision tuning of $α$ is not required for efficient gate compilation, significantly enhancing the physical plausibility of non-semisimple anyonic architectures.

DOI
10.1103/bwpz-4kgt

Related papers

Universal quantum computation using Ising anyons from a non-semisimple topological quantum field theory

Filippo Iulianelli, Sung Kim, Joshua Sussan +1 more

We propose a framework for topological quantum computation using newly discovered non-semisimple analogs of topological quantum field theories in 2 + 1 dimensions. These enhanced theories offer more powerful models for quantum computation. The conventional theory of Ising anyons, which is believed to describe excitations in the ν = 5/2 fractional quantum Hall state, is not universal for quantum computation via braiding of quasiparticles. However, we show that the non-semisimple theory introduces new anyon types that extend the Ising framework. By adding just one new anyon type, universal quantum computation can be achieved through braiding alone. This result opens new avenues for realizing fault-tolerant quantum computing in topologically ordered systems. Ising anyons are a promising platform for topological quantum computation, but braiding alone is not universal. Here, authors show that extending to non-semi simple TQFTs introduces new anyon types that enable universal quantum computation with Ising anyons.

arXiv:2410.14860•Nature Communications•Aug 5, 2025•5 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

Universal Gates from Braiding and Fusing Anyons on Quantum Hardware

Chiu Fan Bowen Lo, Anasuya Lyons, Dan Gresh +8 more

Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, $S_3$, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes $S_3$ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.

quant-pharXiv:2601.20956•Nature•Jan 30, 2026•0 citations

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

Implementing Arbitrary Phase Gates with Ising Anyons

Parsa Bonderson, David J. Clarke, Chetan Nayak +1 more

Ising-type non-Abelian anyons are likely to occur in a number of physical systems, including quantum Hall systems, where recent experiments support their existence. In general, non-Abelian anyons may be utilized to provide a topologically error-protected medium for quantum information processing. However, the topologically protected operations that may be obtained by braiding and measuring topological charge of Ising anyons are precisely the Clifford gates, which are not computationally universal. The Clifford gate set can be made universal by supplementing it with single-qubit pi/8-phase gates. We propose a method of implementing arbitrary single-qubit phase gates for Ising anyons by running a current of anyons with interfering paths around computational anyons.

arXiv:0911.2691•Physical Review Letters•May 7, 2010•67 citations

Fibonacci anyons versus Majorana fermions

Emil Génetay Johansen, Tapio Simula

We have studied ${\rm SU}(2)_k$ anyon models, assessing their prospects for topological quantum computation. In particular, we have compared the Ising ($k=2$) anyon and Fibonacci ($k=3$) anyon models, motivated by their potential for future realizations based on Majorana fermion quasiparticles or exotic fractional quantum-Hall states, respectively. The quantum computational performance of the different anyon models is quantified at single qubit level by the difference between a target unitary operator and its approximation realised by anyon braiding. To facilitate efficient comparisons, we have developed a Monte Carlo enhanced Solovay-Kitaev quantum compiler algorithm that finds optimal braid words in polynomial time from the exponentially large search tree. Since universal quantum computation cannot be achieved within the Ising anyon model by braiding alone, we have introduced an additional elementary phase gate to model a non-topological measurement process, which restores universality of the anyon model at the cost of breaking the full topological protection. We model conventional kinds of decoherence processes algorithmically by introducing a controllable noise term to all non-topological gate operations. We find that for reasonable levels of decoherence, even the hybrid Ising anyon model retains a significant topological advantage over a conventional, non-topological, quantum computer. Furthermore, we find that only surprisingly short anyon braids are ever required to be compiled due to the gate noise exceeding the intrinsic error of the braid words already for word lengths of the order of $100$ elementary braids. We conclude that the future for hybrid topological quantum computation remains promising.

arXiv:2008.10790•Aug 25, 2020•0 citations
Non-semisimple extensions of the Ising anyon model developed in our previous work enable universal topological quantum computation via braiding alone, overcoming the Clifford-only limitation of semisimple theories. The non-semisimple theory provides new anyon types indexed by a real parameter ααα, the neglecton. Braiding acts unitarily with respect to an indefinite Hermitian form, while the computational subspace sits in a positive-definite sector. We demonstrate that this universality is robust, persisting over an open interval of the neglecton parameter ααα where the computational subspace remains positive-definite. We identify special values of ααα where the physical subspace decouples exactly from negative-norm components, ensuring fully unitary evolution and suppressed leakage. We further present an alternative encoding supporting exact single-qubit Clifford gates alongside a non-Clifford phase gate. We show that high-precision tuning of ααα is not required for efficient gate compilation, significantly enhancing the physical plausibility of non-semisimple anyonic architectures.
We propose a framework for topological quantum computation using newly discovered non-semisimple analogs of topological quantum field theories in 2 + 1 dimensions. These enhanced theories offer more powerful models for quantum computation. The conventional theory of Ising anyons, which is believed to describe excitations in the ν = 5/2 fractional quantum Hall state, is not universal for quantum computation via braiding of quasiparticles. However, we show that the non-semisimple theory introduces new anyon types that extend the Ising framework. By adding just one new anyon type, universal quantum computation can be achieved through braiding alone. This result opens new avenues for realizing fault-tolerant quantum computing in topologically ordered systems. Ising anyons are a promising platform for topological quantum computation, but braiding alone is not universal. Here, authors show that extending to non-semi simple TQFTs introduces new anyon types that enable universal quantum computation with Ising anyons.
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.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
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.
Ising-type non-Abelian anyons are likely to occur in a number of physical systems, including quantum Hall systems, where recent experiments support their existence. In general, non-Abelian anyons may be utilized to provide a topologically error-protected medium for quantum information processing. However, the topologically protected operations that may be obtained by braiding and measuring topological charge of Ising anyons are precisely the Clifford gates, which are not computationally universal. The Clifford gate set can be made universal by supplementing it with single-qubit pi/8-phase gates. We propose a method of implementing arbitrary single-qubit phase gates for Ising anyons by running a current of anyons with interfering paths around computational anyons.
We have studied SU(2)k{\rm SU}(2)_kSU(2)k​ anyon models, assessing their prospects for topological quantum computation. In particular, we have compared the Ising (k=2k=2k=2) anyon and Fibonacci (k=3k=3k=3) anyon models, motivated by their potential for future realizations based on Majorana fermion quasiparticles or exotic fractional quantum-Hall states, respectively. The quantum computational performance of the different anyon models is quantified at single qubit level by the difference between a target unitary operator and its approximation realised by anyon braiding. To facilitate efficient comparisons, we have developed a Monte Carlo enhanced Solovay-Kitaev quantum compiler algorithm that finds optimal braid words in polynomial time from the exponentially large search tree. Since universal quantum computation cannot be achieved within the Ising anyon model by braiding alone, we have introduced an additional elementary phase gate to model a non-topological measurement process, which restores universality of the anyon model at the cost of breaking the full topological protection. We model conventional kinds of decoherence processes algorithmically by introducing a controllable noise term to all non-topological gate operations. We find that for reasonable levels of decoherence, even the hybrid Ising anyon model retains a significant topological advantage over a conventional, non-topological, quantum computer. Furthermore, we find that only surprisingly short anyon braids are ever required to be compiled due to the gate noise exceeding the intrinsic error of the braid words already for word lengths of the order of 100100100 elementary braids. We conclude that the future for hybrid topological quantum computation remains promising.