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 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.
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.
FINDING: Non-Abelian anyon fusion algebra exhibits self-similarity under braid group representations, enabling topological quantum computation via braiding operations. MATH: - Braid group Bn generators σi satisfy σiσi+1σi=σi+1σiσi+1 and σiσj=σjσi for ∣i−j∣>1. - Hecke algebra relations: (σi−q1/2)(σi+q−1/2)=0, with q=e2πi/k for SU(2)_k anyons. - Fusion rules: ϕa×ϕb=∑cNabcϕc, where Nabc are non-negative integers (fusion coefficients). - Fibonacci anyon fusion: τ×τ=1+τ, with quantum dimension dτ=ϕ=(1+5)/2≈1.618. CONNECTION: - Fibonacci anyon quantum dimension dτ=ϕ=1.618 (golden ratio). - Braid matrix eigenvalues for Fibonacci anyons: e±3πi/5, related to golden ratio via \( 2\cos(3\pi/5) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
A quantum computer must be capable of manipulating quantum information while at the same time protecting it from error and loss of quantum coherence due to interactions with the environment. Topological quantum computation (TQC) offers a particularly elegant way to achieve this. In TQC, quantum information is stored in exotic states of matter which are intrinsically protected from decoherence, and quantum computation is carried out by dragging particle-like excitations (quasiparticles) around one another in two space dimensions. The resulting quasiparticle trajectories define world-lines in three-dimensional space-time, and the corresponding computation depends only on the topology of the braids formed by the world-lines. Quasiparticles that can be used for TQC are expected to exist in a variety of fractional quantum Hall states, among them the so-called Fibonacci anyons. These quasiparticles are conjectured to exist in the = 12/5 fractional quantum Hall state which has been observed in experiments. It has been shown that qubits can be encoded using three or four Fibonacci anyons and single-qubit gates can be carried out by braiding quasiparticles within each qubit. Braids that approximate single-qubit gates can be found through brute force searching and the result can be systematically improved, to any desired accuracy, by applying the Solovay- Kitaev algorithm in SU(2). Two-qubit gates are significantly harder to implement, mostly due to the following two reasons. First, the Hilbert space of the quasiparticles forming two qubits is considerably larger than the Hilbert space of the quasiparticles of a single qubit. Therefore, performing a brute force search to find braids that approximate two-qubit gates, as well as the implementation of the Solovay-Kitaev algorithm for subsequent improvements are prohibitively more difficult. Second, to construct any entangling two-qubit gate, one needs to braid some xvii of the quasiparticles from one qubit around quasiparticles of the other qubit. This process will inevitably lead to leakage errors, i.e. transitions from the qubit space to other available states in the Hilbert space. In this thesis, I will present several efficient methods to construct two-qubit gates using a specific class of quasiparticles. In particular, I show that the problem of finding braids that correspond to two-qubit gates can be reduced to a series of smaller problems which involve braiding only three objects at a time. The required computational power for finding these braids is equivalent to that needed to find single-qubit gates, therefore, these braids can be found with the same high degree of accuracy and efficiency. The end result of this work is an efficient procedure for translating (or "compiling") arbitrary quantum algorithms into specific braiding patterns for Fibonacci anyons, as well as quasiparticles of certain other fractional quantum Hall states that can be used for TQC.
FINDING: Anyon braiding forms the computational substrate for topological quantum computation, with non-Abelian statistics encoded in braid group representations rather than local qubit states. | MATH: Braid group Bn generators σi satisfy σiσi+1σi=σi+1σiσi+1 (Yang–Baxter equation); non-Abelian anyons carry unitary representations U(σi) with U(σi)U(σj)=U(σj)U(σi) for ∣i−j∣≥2. Fibonacci anyons: fusion rule τ×τ=1+τ, Hilbert space dimension grows as Fibonacci sequence Fn, golden ratio ϕ=(1+5)/2≈1.618 appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-g surfaces: dimension =kg for level-k Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio Fn+1/Fn→1.618 — direct geometric harmony link. Braid group relations ar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Universal Quantum Gate Compilation in SU(2)k Anyon Models via Multiple-Braiding
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∈[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.
Abstract Topological quantum computation is an implementation of a quantum computer in a way that radically reduces decoherence. Topological qubits are encoded in the topological evolution of two-dimensional quasi-particles called anyons and universal set of quantum gates can be constructed by braiding these anyons yielding to a topologically protected circuit model. In the present study we remind the basics of this emerging quantum computation scheme and illustrate how a topological qubit built with three Fibonacci anyons might be adopted to achieve leakage free braiding gate by exchanging the anyons composing it. A single-qubit braiding gate that approximates the Hadamard quantum gate to a certain accuracy is numerically implemented using a brute force search method. The algorithms utilized for that purpose are explained and the numerical programs are publicly shared for reproduction and further use.