We systematically investigate the implementation of a universal gate set via double-braiding within SU(2)k anyon models. The explicit form of the double elementary braiding matrices (DEBMs) in these models are derived from the F-matrices and R-symbols obtained via the q-deformed representation theory of SU(2). Using these EBMs, standard single-qubit gates are synthesized up to a global phase by a Genetic Algorithm-enhanced Solovay-Kitaev Algorithm (GA-enhanced SKA), achieving the accuracy required for fault-tolerant quantum computation with only 2-level decomposition. For two-qubit entangling gates, Genetic Algorithm (GA) yields braidwords of 30 braiding operations that approximate the local equivalence class [CNOT]. Theoretically, we demonstrate that performing double-braiding in a three-anyon (six-anyon) encoding of single-qubit (two-qubit) is topologically equivalent to a protocol requiring the physical manipulation of only one (three) anyons to execute arbitrary braids. Our numerical results provide strong evidence that double-braiding in SU(2)k anyons models is capable of universal quantum computation. Moreover, the proposed protocol offers a potential new strategy for significantly reducing the number of non-Abelian anyons that need to be physically manipulated in future braiding-based topological quantum computations (TQC).
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∈[2.001,2.022], a few braiding operations can approximate the local equivalence class
[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.
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.
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,
S3, 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
S3 TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
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′, 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} 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}. The obtained accuracy of the \textit{H}/\textit{T}-gate by
{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,
V3131 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} 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
V3113.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.
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.
We have studied
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.