Recent experiments have focused attention on the properties of chains of atoms in which the atoms are either in their ground states or in highly excited Rydberg states which block similar excitations in their immediate neighbors. As the low energy Hilbert space of such chains is isomorphic to that of a chain of Fibonacci anyons, they have been proposed as a platform for topological quantum computation and for simulating anyon dynamics. We show that generic local operators in the Rydberg chain correspond to non-local anyonic operators that do not preserve a topological symmetry of the physical anyons. Consequently, we argue that Rydberg chains do not yield Fibonacci anyons and quantum computation with Rydberg atoms is not topologically protected.
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 …
A defining property of particles is their behavior under exchange. In two dimensions anyons can exist which, opposed to fermions and bosons, gain arbitrary relative phase factors or even undergo a change of their type. In the latter case one speaks of non-Abelian anyons - a particularly simple and aesthetic example of which are Fibonacci anyons. They have been studied in the context of fractional quantum Hall physics where they occur as quasiparticles in the $k=3$ Read-Rezayi state, which is conjectured to describe a fractional quantum Hall state at filling fraction $ν=12/5$. Here we show that…
Anyonic states, which are topologically robust originated from their peculiar structure of Hilbert space, have important applications in quantum computing and quantum communication. When an anyonic state is used as an information carrier of the deterministic quantum one-time pad (DQOTP), we find that the Fibonacci particle-antiparticle pair produced from vacuum can be used to asymptotically send $2\log_2 d_τ$ bits of classical information ($d_τ$ is the quantum dimension of a Fibonacci anyon $τ$), which equals to the anyonic mutual information of the pair. Furthermore, by studying the DQOTP via…
Phases of Interacting Fibonacci Anyons on a Ladder at Half-Filling
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；University of Nottingham（诺丁汉大学）
Topological quantum computation (TQC) is one of the most striking architectures that can realize fault-tolerant quantum computers. In TQC, the logical space and the quantum gates are topologically protected, i.e., robust against local disturbances. The topological protection, however, requires rather complicated lattice models and hard-to-manipulate dynamics; even the simplest system that can realize universal TQC--the Fibonacci anyon system--lacks a physical realization, let alone braiding the non-Abelian anyons. Here, we propose a disk model that can realize the Fibonacci anyon system, and c…
Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibo…
The emergence of non-Abelian anyons from large collections of interacting elementary particles is a conceptually beautiful phenomenon with important ramifications for fault-tolerant quantum computing. Over the last few decades the field has evolved from a highly theoretical subject to an active experimental area, particularly following proposals for trapping non-Abelian anyons in `engineered' structures built from well-understood components. In this short overview we briefly tour the impressive progress that has taken place in the quest for the simplest type of non-Abelian anyon---defects bind…
Topological quantum computation with Fibonacci anyons relies on the possibility of efficiently generating unitary transformations upon pseudoparticles braiding. The crucial fact that such set of braids has a dense image in the unitary operations space is well known; in addition, the Solovay-Kitaev algorithm allows to approach a given unitary operation to any desired accuracy. In this paper, the latter task is fulfilled with an alternative method, in the SU(2) case, based on a generalization of the geodesic dome construction to higher dimension.
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.
Classically simulating the dynamics of anyonic excitations in two-dimensional quantum systems is likely intractable in general because such dynamics are sufficient to implement universal quantum computation. However, processes of interest for the study of quantum error correction in anyon systems are typically drawn from a restricted class that displays significant structure over a wide range of system parameters. We exploit this structure to classically simulate, and thereby demonstrate the success of, an error-correction protocol for a quantum memory based on the universal Fibonacci anyon mo…
Quantum spin models have been studied extensively in one and higher dimensions. Furthermore, these systems have been doped with holes to study $t$--$J$ models of $SU(2)$ spin-1/2. Their anyonic counterparts can be built from non-Abelian anyons, such as Fibonacci anyons described by $SU(2)_3$ theories, which are quantum deformations of the $SU(2)$ algebra. Inspired by the physics of $SU(2)$ spins, several works have explored ladders of Fibonacci anyons and also one-dimensional (1D) $t$--$J$ models. Here we aim to explore the combined effects of extended dimensionality and doping by studying lad…
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 hav…
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 …
A topological quantum computer should allow intrinsically fault-tolerant quantum computation, but there remains uncertainty about how such a computer can be implemented. It is known that topological quantum computation can be implemented with limited quasiparticle braiding capabilities, in fact using only a single mobile quasiparticle, if the system can be properly initialized by measurements. It is also known that measurements alone suffice without any braiding, provided that the measurement devices can be dynamically created and modified. We study a model in which both measurement and braidi…
We show that quasicrystals exhibit anyonic behavior that can be used for topological quantum computing. In particular, we study a correspondence between the fusion Hilbert spaces of the simplest non-abelian anyon, the Fibonacci anyons, and the tiling spaces of a class of quasicrystals, which includes the one dimensional Fibonacci chain and the two dimensional Penrose tiling. A possible encoding on tiling spaces of topological quantum information processing is also discussed.
We study the reaction-diffusion dynamics of Fibonacci anyons in a one dimensional lattice. Due to their non-Abelian nature, besides the position degree of freedom (DOF), these anyons also have a nonlocal internal DOF, which can be characterized by a fusion tree. We first consider a pure-reaction dynamics associated with the internal DOF, which is of intrinsically quantum origin, with either an "all-$τ$" or "completely random" initial fusion tree. These two fusion trees are unstable and likely stable steady states for the internal DOF, respectively. We obtain the decay rate of the anyon number …
We consider a hypothetical topological quantum computer where the qubits are comprised of either Ising or Fibonacci anyons. For each case, we calculate the time and number of qubits (space) necessary to execute the most computationally expensive step of Shor's algorithm, modular exponentiation. For Ising anyons, we apply Bravyi's distillation method [S. Bravyi, Phys. Rev. A 73, 042313 (2006)] which combines topological and non-topological operations to allow for universal quantum computation. With reasonable restrictions on the physical parameters we find that factoring a 128 bit number requir…
Anyons exist as point like particles in two dimensions and carry braid statistics which enable interactions that are independent of the distance between the particles. Except for a relatively few number of models which are analytically tractable, much of the physics of anyons remain still unexplored. In this paper, we show how U(1)-symmetry can be combined with the previously proposed anyonic Matrix Product States to simulate ground states and dynamics of anyonic systems on a lattice at any rational particle number density. We provide proof of principle by studying itinerant anyons on a one di…
We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial (`identity') channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent z=1, and described by a two-dimensional conformal field theory with central charge c=7/10. An exact mapping of the anyonic chain onto the two-dimensional tricritical Ising…
The content of this thesis can be broadly summarised into two categories: first, I constructed modified numerical algorithms based on tensor networks to simulate systems of anyons in low dimensions, and second, I used those methods to study the topological phases the anyons form when they braid around one another. In the first phase of my thesis, I extended the anyonic tensor network algorithms, by incorporating U(1) symmetry to give a modified ansatz, Anyon-U(1) tensor networks, which are capable of simulating anyonic systems at any rational filling fraction. In the second phase, I used the n…
The golden chain with antiferromagnetic interaction is an anyonic system of particular interest as when all anyons are confined to the chain, it is readily stabilised against fluctuations away from criticality. However, additional local scaling operators have recently been identified on the disc which may give rise to relevant fluctuations in the presence of free charges. Motivated by these results for Fibonacci anyons, this paper presents a systematic method of identifying all topological sectors of local scaling operators for critical anyon rings of arbitrary winding number on surfaces of ar…
Many phases of matter, including superconductors, fractional quantum Hall fluids and spin liquids, are described by gauge theories with constrained Hilbert spaces. However, thermalization and the applicability of quantum statistical mechanics has primarily been studied in unconstrained Hilbert spaces. In this article, we investigate whether constrained Hilbert spaces permit local thermalization. Specifically, we explore whether the eigenstate thermalization hypothesis (ETH) holds in a pinned Fibonacci anyon chain, which serves as a representative case study. We first establish that the constra…
Using finite size scaling arguments, the critical properties of a chain of interacting anyons can be extracted from the low energy spectrum of a finite system. In Phys. Rev. Lett. 98, 160409 (2007), Feiguin et al. showed that an antiferromagnetic (AFM) chain of Fibonacci anyons on a torus is in the same universality class as the tricritical Ising model, and that criticality is protected by a topological symmetry. In the present paper we first review the graphical formalism for the study of anyons on the disc and demonstrate how this formalism may be consistently extended to the study of system…
In a topological quantum computer, universality is achieved by braiding and quantum information is natively protected from small local errors. We address the problem of compiling single-qubit quantum operations into braid representations for non-abelian quasiparticles described by the Fibonacci anyon model. We develop a probabilistically polynomial algorithm that outputs a braid pattern to approximate a given single-qubit unitary to a desired precision. We also classify the single-qubit unitaries that can be implemented exactly by a Fibonacci anyon braid pattern and present an efficient algori…
Quantum gates built out of braid group elements form the building blocks of topological quantum computation. They have been extensively studied in $SU(2)_k$ quantum group theories, a rich source of examples of non-Abelian anyons such as the Ising ($k=2$), Fibonacci ($k=3$) and Jones-Kauffman ($k=4$) anyons. We show that the fusion spaces of these anyonic systems can be precisely mapped to the product state zero modes of certain Nicolai-like supersymmetric spin chains. As a result, we can realize the braid group on the product state zero modes of these supersymmetric systems. These operators ki…
Read-Rezayi fractional quantum Hall states are among the prime candidates for realizing non-Abelian anyons which in principle can be used for topological quantum computation. We present a prescription for efficiently finding braids which can be used to carry out a universal set of quantum gates on encoded qubits based on anyons of the Read-Rezayi states with $k>2$, $k\neq4$. This work extends previous results which only applied to the case $k = 3$ (Fibonacci) and clarifies why in that case gate constructions are simpler than for a generic Read-Rezayi state.
We describe a method for engineering local $k+1$-body interactions ($k=1,2,3$) from two-body couplings in spin-${1}{2}$ systems. When implemented in certain systems with a flat single-particle band with a unit Chern number, the resulting many-body ground states are fractional Chern insulators which exhibit abelian and non-abelian anyon excitations. The most complex of these, with $k=3$, has Fibonacci anyon excitations; our system is thus capable of universal topological quantum computation. We then demonstrate that an appropriately tuned circuit of qubits could faithfully replicate this model …
Majorana-based quantum gates are not complete for performing universal topological quantum computation while Fibonacci-based gates are difficult to be realized electronically and hardly coincide with the conventional quantum circuit models. In Ref. \cite{hukane}, it has been shown that a strongly correlated Majorana edge mode in a chiral topological superconductor can be decomposed into a Fibobacci anyon $τ$ and a thermal operator anyon $\varepsilon$ in the tricritical Ising model. The deconfinement of $τ$ and $\varepsilon$ via the interaction between the fermion modes yields the anyon {collis…
