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arXiv:1802.06176 · Quantum Science and Technology

Introduction to topological quantum computation with non-Abelian anyons

Bernard Field, Tapio Simula

Published Jun 6, 2018• 135 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/2058-9565/aacad2

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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.
Quantum Computing with Z2\mathbb{Z}_2Z2​ Abelian anyon system
Topological quantum computers provide a fault-tolerant method for performing quantum computation. Topological quantum computers manipulate topological defects with exotic exchange statistics called anyons. The simplest anyon model for universal topological quantum computation is the Fibonacci anyon model, which is a non-abelian anyon system. In non-abelian anyon systems, exchanging anyons always results a unitary operations instead of a simple phase changing in abelian anyon systems. So, non-abelian anyon systems are of the interest to topological quantum computation. Up till now, most people still hold the belief that topological quantum computions can be implemented only on the non-abelian anyon systems. But actually this is not true. Inspired by extrinsic semiconductor technology, we suggest that abelian anyon systems with defects also support topological quantum computing. In this letter, we report a topological quantum computer prototype based on Z2\mathbb{Z}_2Z2​ abelian anyon system.
Anyons are quasiparticles that may be realized in two dimensional systems. They come in two types, the simpler Abelian anyons and the more complex non-Abelian anyons. Both of these have been considered as a means for quantum computation, but non-Abelian anyons are usually assumed to be better suited to the task. Here we challenge this view, demonstrating that Abelian anyon models have as much potential as some simple non-Abelian models. First the means to perform quantum computation with Abelian anyon models is considered. These models, like many non-Abelian models, cannot realize universal quantum computation by braiding alone. Non-topological operations must be used in addition, whose complexity depends on the physical means by which the anyons are realized. Here we consider anyons based on spin lattice models, with single spin measurements playing the role of non-topological operations. The computational power achieved by various kinds of measurement is explored and the requirements for universality are determined. The possibility to simulate non-Abelian anyons using Abelian ones is then considered. Finally, a non-Abelian quantum memory is dissected in order to determine the means by which it provides fault-tolerant storage of information. This understanding is then employed to build equivalent quantum memories with Abelian anyon models. The methodology provides with the means to demonstrate that Abelian models have the capability to simulate non-Abelian anyons, and to realize the same computational power and fault-tolerance as non Abelian models. Apart from the intellectual interest in relating topological models with each other, and of understanding the properties of non-Abelian anyons in terms of the simpler Abelian ones, these results can also be applied in the lab. The simpler structure of Abelian anyons means that their physical realization is more straightforward. The demonstration of non-Abelian properties with Abelian models therefore allows features of non-Abelian anyons to be realized with present and near future technology. Based on this possibility, proposals are made here for proof of principle experiments.
FINDING: Topological quantum computing uses non-Abelian anyons (Majorana zero modes) for fault-tolerant qubits, with braiding operations encoding quantum gates. | MATH: Braid group Bn B_n Bn​ generators 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​; Majorana operators γi \gamma_i γi​ satisfy {γi,γj}=2δij \{\gamma_i, \gamma_j\} = 2\delta_{ij} {γi​,γj​}=2δij​; fusion rules for Ising anyons: σ×σ=1+ψ \sigma \times \sigma = 1 + \psi σ×σ=1+ψ, σ×ψ=σ \sigma \times \psi = \sigma σ×ψ=σ, ψ×ψ=1 \psi \times \psi = 1 ψ×ψ=1. | CONNECTION: Braid group is deeply linked to the golden ratio via the Temperley-Lieb algebra at q=eiπ/5 q = e^{i\pi/5} q=eiπ/5, where the Jones polynomial yields τ=(1+5)/2≈1.618 \tau = (1+\sqrt{5})/2 \approx 1.618 τ=(1+5​)/2≈1.618; Fibonacci anyons have fusion rules τ×τ=1+τ \tau \times \tau = 1 + \tau τ×τ=1+τ, directly encoding the golden ratio. | DEPTH: 9 — Non-Abelian statistics and braiding are foundational to topological quantum computation, with direct geometric harmony in Fibonacci anyon models. 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.
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.