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arXiv:1511.00719 · Physical Review A

Systematically generated two-qubit anyon braids

Caitlin Carnahan, Daniel Zeuch, N. E. Bonesteel

Published May 20, 2016• 13 citations
arXivHTMLPDF

Abstract

Fibonacci anyons are non-Abelian particles for which braiding is universal for quantum computation. Reichardt has shown how to systematically generate nontrivial braids for three Fibonacci anyons which yield unitary operations with off-diagonal matrix elements that can be made arbitrarily small in a particular natural basis through a simple and efficient iterative procedure. This procedure does not require brute force search, the Solovay-Kitaev method, or any other numerical technique, but the phases of the resulting diagonal matrix elements cannot be directly controlled. We show that despite this lack of control the resulting braids can be used to systematically construct entangling gates for two qubits encoded by Fibonacci anyons.

DOI
10.1103/physreva.93.052328

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FINDING: Topological quantum computation via non-Abelian anyon braiding, specifically Fibonacci anyons, encodes qubits in degenerate state spaces where braid operations generate the Fibonacci fusion rules and yield universal quantum gates. | MATH: Fibonacci anyon fusion rule: τ ⊗ τ = 1 ⊕ τ (golden anyon). Dimension of Hilbert space for n anyons grows as Fibonacci numbers: dim(H_n) = F_{n-1} (with F_0=0, F_1=1). Braid generators σ_i satisfy Yang-Baxter: σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1}, and σ_i σ_j = σ_j σ_i for |i-j|≥2. Single-qubit braiding gates compiled via brute-force optimization (arXiv:2008.03542) — the key constant is the golden ratio φ = (1+√5)/2 ≈ 1.618, which appears as the quantum dimension d_τ = φ. | CONNECTION: Direct geometric harmony: Fibonacci anyons have quantum dimension exactly φ = 1.618 (and its inverse 0.618 = 1/φ). The fusion space dimension sequence (1,1,2,3,5,8,...) is the Fibonacci sequence — the same ratio governing pentagonal symmetry (5-fold crystallogr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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•Jan 1, 2007•0 citations
Fibonacci anyons are non-Abelian particles for which braiding is universal for quantum computation. Reichardt has shown how to systematically generate nontrivial braids for three Fibonacci anyons which yield unitary operations with off-diagonal matrix elements that can be made arbitrarily small in a particular natural basis through a simple and efficient iterative procedure. This procedure does not require brute force search, the Solovay-Kitaev method, or any other numerical technique, but the phases of the resulting diagonal matrix elements cannot be directly controlled. We show that despite this lack of control the resulting braids can be used to systematically construct entangling gates for two qubits encoded by Fibonacci anyons.
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.
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.
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.
Fibonacci anyons are attractive for use in topological quantum computation because any unitary transformation of their state space can be approximated arbitrarily accurately by braiding. However there is no known braid that entangles two qubits without leaving the space spanned by the two qubits. In other words, there is no known "leakage-free" entangling gate made by braiding. In this paper, we provide a remedy to this problem by supplementing braiding with measurement operations in order to produce an exact controlled rotation gate on two qubits.
FINDING: Topological quantum computation via non-Abelian anyon braiding, specifically Fibonacci anyons, encodes qubits in degenerate state spaces where braid operations generate the Fibonacci fusion rules and yield universal quantum gates. | MATH: Fibonacci anyon fusion rule: τ ⊗ τ = 1 ⊕ τ (golden anyon). Dimension of Hilbert space for n anyons grows as Fibonacci numbers: dim(H_n) = F_{n-1} (with F_0=0, F_1=1). Braid generators σ_i satisfy Yang-Baxter: σ_i σ_{i+1} σ_i = σ_{i+1} σ_i σ_{i+1}, and σ_i σ_j = σ_j σ_i for |i-j|≥2. Single-qubit braiding gates compiled via brute-force optimization (arXiv:2008.03542) — the key constant is the golden ratio φ = (1+√5)/2 ≈ 1.618, which appears as the quantum dimension d_τ = φ. | CONNECTION: Direct geometric harmony: Fibonacci anyons have quantum dimension exactly φ = 1.618 (and its inverse 0.618 = 1/φ). The fusion space dimension sequence (1,1,2,3,5,8,...) is the Fibonacci sequence — the same ratio governing pentagonal symmetry (5-fold crystallogr 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.