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Computing with Anyons

Steven H. Simon

Published Sep 29, 2023• 0 citations
Publisher

Abstract

Abstract We introduce some basic ideas of quantum computing including the idea of a quantum circuit model and the idea of a universal quantum computer — implementing arbitrary gates arbitrarily accurately. We discuss how braiding of certain anyons can be used as a universal quantum computer, and we illustrate this using Fibonacci anyons as an example. We then turn to the task of designing braids to perform particular computations, which is known as Topological Quantum compiling.

DOI
10.1093/oso/9780198886723.003.0011

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Abstract We introduce some basic ideas of quantum computing including the idea of a quantum circuit model and the idea of a universal quantum computer — implementing arbitrary gates arbitrarily accurately. We discuss how braiding of certain anyons can be used as a universal quantum computer, and we illustrate this using Fibonacci anyons as an example. We then turn to the task of designing braids to perform particular computations, which is known as Topological Quantum compiling.
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 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 Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.
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 Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots.
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.
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.