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arXiv:0812.2414 · Physical Review A

Exploiting geometric degrees of freedom in topological quantum computing

Haitan Xu, Xin Wan

Published Jul 8, 2009• 12 citations
arXivHTMLPDF

Abstract

In a topological quantum computer, braids of non-Abelian anyons in a $(2+1)$-dimensional space time form quantum gates, whose fault tolerance relies on the topological, rather than geometric, properties of the braids. Here we propose to create and exploit redundant geometric degrees of freedom to improve the theoretical accuracy of topological single- and two-qubit quantum gates. We demonstrate the power of the idea using explicit constructions in the Fibonacci model. We compare its efficiency with that of the Solovay-Kitaev algorithm and explain its connection to the leakage errors reduction in an earlier construction [H. Xu and X. Wan, Phys. Rev. A 78, 042325 (2008)].

DOI
10.1103/physreva.80.012306

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In a topological quantum computer, braids of non-Abelian anyons in a (2+1)(2+1)(2+1)-dimensional space time form quantum gates, whose fault tolerance relies on the topological, rather than geometric, properties of the braids. Here we propose to create and exploit redundant geometric degrees of freedom to improve the theoretical accuracy of topological single- and two-qubit quantum gates. We demonstrate the power of the idea using explicit constructions in the Fibonacci model. We compare its efficiency with that of the Solovay-Kitaev algorithm and explain its connection to the leakage errors reduction in an earlier construction [H. Xu and X. Wan, Phys. Rev. A 78, 042325 (2008)].
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.
A central problem in realizing universal quantum computers is dealing with decoherence effects, which can destroy the vulnerable quantum states present in the internal registers. Several schemes have been developed that provide resistance to decoherence, leading to theories of fault-tolerant quantum computation. One such scheme is the theory of topological quantum computation, which encodes qubit states in robust topological properties of certain quantum systems. However, fault-tolerant theories typically only provide a finite set of primitive gates, from which we must then form approximations to arbitrary quantum gates. The Solovay-Kitaev algorithm describes a procedure for accomplishing this, and proves that the number of primitive gates required to approximate an arbitrary gate is only polylogarithmic in the desired degree of accuracy. In this thesis, we first outline the basic theory of quantum computation. Secondly, we describe the general problem of quantum compilation, list several possible strategies for relevant optimizations, and outline a proof of the Solovay-Kitaev theorem. Next, we give an introduction to the theory of topological quantum computation by anyons, with special focus on Fibonacci-anyons. Finally, we list some numerical results of an implementation of the Solovay-Kitaev algorithm when applied to a Fibonacci-anyon model and analyze the performance gains obtained through applications of the proposed optimizations.
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.
Abstract A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this paper, the Hilbert space for a topological qubit is discussed. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. The mathematical model, for the fusion and braiding matrices of metaplectic anyons, is the quantum deformation of the recoupling theory. We proposed that the existing quantum ternary arithmetic gates can be realized by braiding and topological charge measurement of the metaplectic anyons.
Topological quantum computation with Fibonacci anyons relies on the possibility of efficiently generating unitary transformations upon pseudoparticles braiding. The crucial fact that such a set of braids has a dense image in the unitary operations space is well known; in addition, the Solovay–Kitaev algorithm allows us 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.