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arXiv:1506.03815 · Physical Review A

Classical simulation of quantum error correction in a Fibonacci anyon code

Simon Burton, Courtney G Brell, Steven T. Flammia

Published Feb 7, 2017• 21 citations
arXivHTMLPDF

Abstract

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 model. We numerically simulate a phenomenological model of the system and noise processes on lattice sizes of up to $128\ifmmode\times\else\texttimes\fi{}128$ sites, and find a lower bound on the error-correction threshold of approximately 0.125 errors per edge, which is comparable to those previously known for Abelian and (nonuniversal) non-Abelian anyon models.

DOI
10.1103/physreva.95.022309

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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 model. We numerically simulate a phenomenological model of the system and noise processes on lattice sizes of up to 128\ifmmode×\else\texttimes\fi128128\ifmmode\times\else\texttimes\fi{}128128\ifmmode×\else\texttimes\fi128 sites, and find a lower bound on the error-correction threshold of approximately 0.125 errors per edge, which is comparable to those previously known for Abelian and (nonuniversal) non-Abelian anyon models.
We consider a two-dimensional quantum memory of qubits on a torus which encode the extended Fibonacci string-net code and devise strategies for error correction when those qubits are subjected to depolarizing noise.Building on the concept of tube algebras, we construct a set of measurements and of quantum gates which map arbitrary qubit errors to the string-net subspace and allow for the characterization of the resulting error syndrome in terms of doubled Fibonacci anyons.Tensor network techniques then allow us to quantitatively study the action of Pauli noise on the string-net subspace.We perform Monte Carlo simulations of error correction in this Fibonacci code and compare the performance of several decoders.For the case of a fixed-rate sampling depolarizing noise model, we find an error correction threshold of 4.7% using a clustering decoder.
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.
We consider two-dimensional lattice models that support Ising anyonic excitations and are coupled to a thermal bath.We propose a phenomenological model for the resulting short-time dynamics that includes pair creation, hopping, braiding, and fusion of anyons.By explicitly constructing topological quantum error-correcting codes for this class of system, we use our thermalization model to estimate the lifetime of the quantum information stored in the encoded spaces.To decode and correct errors in these codes, we adapt several existing topological decoders to the non-Abelian setting.We perform large-scale numerical simulations of these two-dimensional Ising anyon systems and find that the thresholds of these models range from 13% to 25%.To our knowledge, these are the first numerical threshold estimates for quantum codes without explicit additive structure.
Quantum gates in topological quantum computation are performed by braiding non-Abelian anyons. These braiding processes can presumably be performed with very low error rates. However, to make a topological quantum computation architecture truly scalable, even rare errors need to be corrected. Error correction for non-Abelian anyons is complicated by the fact that it needs to be performed on a continuous basis, and further errors may occur while we are correcting existing ones. Here, we prove the feasibility of this task, establishing non-Abelian anyons as a viable platform for scalable quantum computation. We thereby focus on Ising anyons as the most prominent example of non-Abelian anyons and show that for these a finite error rate can indeed be corrected continuously. There is a threshold error rate pc>0{p}_{c}>0pc​>0 such that for all error rates p<pcp<{p}_{c}p<pc​ the probability of a logical error per time step can be made exponentially small in the distance of a logical qubit.
Like their classical counterparts, quantum codes are designed to pro- tect quantum in- formation from noise. From the perspective of informa- tion theory one considers the op- erations required to restore the encoded information given a syndrome which diagnoses the noise. From a more physics perspective, one considers systems whose energetically protected groundspace encodes the information. In this work we show that standard error correction procedures can be applied to systems where the noise ap- pears as non- abelian Fibonacci anyons. In the case of a Hamiltonian with non-commuting terms, we build a theory describing the spectrum of these models, with particular focus on the 3D gauge color code model. Numerics support the conjecture that this model is gapped, which one would expect for a self-correcting quantum memory.
The simulation of non-Abelian anyon braiding is a critical step towards fault-tolerant quantum computation. We introduce a framework for this task based on a one-dimensional Quasicrystal Inflation Code (QIC). The code is defined by a local Hamiltonian whose ground state manifold enforces Fibonacci tiling constraints and possesses the correct Fibonacci degeneracy. We derive the corresponding braid operators and demonstrate that, while they are formally non-local, they possess an exact, local 3-qubit structure. This allows us to distill their action into a single, physically realizable 8×88 \times 88×8 gate, which we term the BgateB_{gate}Bgate​. We prove through comparative compilation for systems with different numbers of qubits that constructing circuits by composing these local gates is dramatically more scalable than compiling the equivalent global unitary, showing a greater than tenfold reduction in circuit depth. To validate the framework, we successfully executed a braiding algorithm for the Jones polynomial, a topological invariant of knots, on an IBM Quantum processor, quantifying the signal degradation due to hardware noise. Finally, we numerically confirm for systems up to 17 qubits that our construction rigorously satisfies the required Temperley-Lieb and braid group relations while preserving the code subspace. This work establishes a validated and physically grounded pathway for the scalable simulation of anyonic braiding on programmable quantum systems.