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arXiv:2404.01779

Diagonal Coset Approach to Topological Quantum Computation with Fibonacci Anyons

Lachezar S. Georgiev, Ludmil K. Hadjiivanov, Grigori Matein

Published Apr 2, 2024• 0 citations
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Abstract

We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the $\mathbb{Z}_3$ parafermion fractional quantum Hall state in the second Landau level with filling factor $ν=12/5$. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the $n$-qubit topological quantum register in terms of $2n+2$ Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in $\mathbb{Z}_3$ parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.

DOI
10.48550/arxiv.2404.01779

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We investigate a promising conformal field theory realization scheme for topological quantum computation based on the Fibonacci anyons, which are believed to be realized as quasiparticle excitations in the Z3\mathbb{Z}_3Z3​ parafermion fractional quantum Hall state in the second Landau level with filling factor ν=12/5ν=12/5ν=12/5. These anyons are non-Abelian and are known to be capable of universal topological quantum computation. The quantum information is encoded in the fusion channels of pairs of such non-Abelian anyons and is protected from noise and decoherence by the topological properties of these systems.The quantum gates are realized by braiding of these anyons. We propose here an implementation of the nnn-qubit topological quantum register in terms of 2n+22n+22n+2 Fibonacci anyons. The matrices emerging from the anyon exchanges, i.e. the generators of the braid group for one qubit are derived from the coordinate wave functions of a large number of electron holes and 4 Fibonacci anyons which can furthermore be represented as correlation functions in Z3\mathbb{Z}_3Z3​ parafermionic two-dimensional conformal field theory. The representations of the braid groups for more than 4 anyons are obtained by fusing pairs of anyons before braiding, thus reducing eventually the system to 4 anyons.
A bstract Fibonacci anyons ε provide the simplest possible model of non-Abelian fusion rules: [1] × [1] = [0] ⊕ [1]. We propose a conformal field theory construction of topological quantum registers based on Fibonacci anyons realized as quasiparticle excitations in the ℤ 3 parafermion fractional quantum Hall state. To this end, the results of Ardonne and Schoutens for the correlation function of four Fibonacci fields are extended to the case of arbitrary number n of quasi-holes and N = 3 r electrons. Special attention is paid to the braiding properties of the obtained correlators. We explain in details the construction of a monodromy representation of the Artin braid group B \mathcal{B} B B n acting on n -point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n . Finally, we construct N \mathcal{N} N N qubit computational spaces in terms of conformal blocks of 2N 2\mathcal{N} 2N 2 N + 2 Fibonacci 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.
Abstract Quantum many-body systems with a non-Abelian topological order can host anyonic quasiparticles. It has been proposed that anyons could be used to encode and manipulate information in a topologically protected manner that is immune to local noise, with quantum gates performed by braiding and fusing anyons. Unfortunately, realizing non-Abelian topologically ordered states is challenging, and it was not until recently that the signatures of non-Abelian statistics were observed through digital quantum simulation approaches. However, not all forms of topological order can be used to realize universal quantum computation. Here we use a superconducting quantum processor to simulate non-Abelian topologically ordered states of the Fibonacci string-net model and demonstrate braidings of Fibonacci anyons featuring universal computational power. We demonstrate the non-trivial topological nature of the quantum states by measuring the topological entanglement entropy. In addition, we create two pairs of Fibonacci anyons and demonstrate their fusion rule and non-Abelian braiding statistics by applying unitary gates on the underlying physical qubits. Our results establish a digital approach to explore non-Abelian topological states and their associated braiding statistics with current noisy intermediate-scale quantum processors.
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 BnB_nBn​ generators σi\sigma_iσi​ satisfy σiσi+1σi=σi+1σiσi+1\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}σi​σi+1​σi​=σi+1​σi​σi+1​ (Yang–Baxter equation); non-Abelian anyons carry unitary representations U(σi)U(\sigma_i)U(σi​) with U(σi)U(σj)=U(σj)U(σi)U(\sigma_i)U(\sigma_j)=U(\sigma_j)U(\sigma_i)U(σi​)U(σj​)=U(σj​)U(σi​) for ∣i−j∣≥2|i-j|\geq 2∣i−j∣≥2. Fibonacci anyons: fusion rule τ×τ=1+τ\tau \times \tau = 1 + \tauτ×τ=1+τ, Hilbert space dimension grows as Fibonacci sequence FnF_nFn​, golden ratio ϕ=(1+5)/2≈1.618\phi = (1+\sqrt{5})/2 \approx 1.618ϕ=(1+5​)/2≈1.618 appears as asymptotic dimension growth rate. Topological protection arises from degeneracy of ground states on genus-ggg surfaces: dimension =kg= k^g=kg for level-kkk Chern–Simons theory. | CONNECTION: The Fibonacci anyon fusion space dimension ratio Fn+1/Fn→1.618F_{n+1}/F_n \to 1.618Fn+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
FINDING: Anyon topological qubits rely on braiding non-Abelian anyons in 2D systems, with flux qubits enabling readout via Josephson vortices, and Fibonacci anyons providing universal gate sets through braiding operations. MATH: Braiding group generators σi \sigma_i σi​ satisfy σiσi+1σi=σi+1σiσi+1 \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} σi​σi+1​σi​=σi+1​σi​σi+1​ (Yang–Baxter equation). Fibonacci anyon fusion rules: τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ, with quantum dimension dτ=ϕ=(1+5)/2≈1.618 d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618 dτ​=ϕ=(1+5​)/2≈1.618. Single-qubit braiding gates compiled from Fibonacci anyons use matrices with entries involving ϕ−1≈0.618 \phi^{-1} \approx 0.618 ϕ−1≈0.618. Chern–Simons term in Ginzburg–Landau model: SCS=k4π∫A∧dA S_{CS} = \frac{k}{4\pi} \int A \wedge dA SCS​=4πk​∫A∧dA, with level k k k integer. CONNECTION: Fibonacci anyon quantum dimension dτ=ϕ d_\tau = \phi dτ​=ϕ directly ties to the golden ratio 1.618 and its inverse 0.618. Braiding matrices for Fibonacci anyons involve ϕ−1 \phi^{-1} ϕ−1 and ϕ−2≈0.382 \phi^{-2} \approx 0.382 ϕ−2≈0.382. The Yan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.