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arXiv:2012.04610 · Physical Review X

Quantum Error Correction Thresholds for the Universal Fibonacci Turaev-Viro Code

Alexis Schotte, Guanyu Zhu, Lander Burgelman, Frank Verstraete

Published Apr 15, 2022• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevx.12.021012

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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.
FINDING: Quantum error correction thresholds are noise-rate bounds, not golden-ratio constants; the Fibonacci Turaev-Viro code links topological order to Fibonacci numbers, but no direct golden-ratio threshold emerges from these abstracts. | MATH: Threshold theorems (e.g., ϵth>0\epsilon_{\text{th}} > 0ϵth​>0 for fault tolerance); Pauli noise estimation via process tomography (Kliesch: sample complexity O(n2/ϵ2)\mathcal{O}(n^2/\epsilon^2)O(n2/ϵ2) for nnn-qubit noise channels); Fibonacci Turaev-Viro code: anyon fusion rules 1×τ=τ1 \times \tau = \tau1×τ=τ, τ×τ=1+τ\tau \times \tau = 1 + \tauτ×τ=1+τ, with quantum dimension dτ=ϕ=(1+5)/2≈1.618d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618dτ​=ϕ=(1+5​)/2≈1.618; Rydberg erasure thresholds: biased noise reduces effective threshold by factor η=perasure/ptotal\eta = p_{\text{erasure}}/p_{\text{total}}η=perasure​/ptotal​. | CONNECTION: Fibonacci anyons have quantum dimension ϕ=1.618\phi = 1.618ϕ=1.618 — the golden ratio appears as a *topological charge dimension*, not as a threshold value. The Turaev-Viro invariant is a state-sum over triangulations, linking Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Quantum error correction is instrumental in protecting quantum systems from noise in quantum computing and communication settings. Pauli channels can be efficiently simulated and threshold values for Pauli error rates under a variety of error-correcting codes have been obtained. However, realistic quantum systems can undergo noise processes that differ significantly from Pauli noise. In this paper, we present an efficient hard decoding algorithm for optimizing thresholds and lowering failure rates of an error-correcting code under general completely positive and trace-preserving (i.e., Markovian) noise. We use our hard decoding algorithm to study the performance of several error-correcting codes under various non-Pauli noise models by computing threshold values and failure rates for these codes. We compare the performance of our hard decoding algorithm to decoders optimized for depolarizing noise and show improvements in thresholds and reductions in failure rates by several orders of magnitude. Our hard decoding algorithm can also be adapted to take advantage of a code's non-Pauli transversal gates to further suppress noise. For example, we show that using the transversal gates of the 5-qubit code allows arbitrary rotations around certain axes to be perfectly corrected. Furthermore, we show that Pauli twirling can increase or decrease the threshold depending upon the code properties. Lastly, we show that even if the physical noise model differs slightly from the hypothesized noise model used to determine an optimized decoder, failure rates can still be reduced by applying our hard decoding algorithm.
A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. Here we demonstrate that non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved over a distance of order the code distance, and thus braided, by a constant depth local unitary quantum circuit followed by a permutation of qubits. Our gates are protected in the sense that the length of error strings do not grow by more than a constant factor. When applied to the Fibonacci surface code, our results demonstrate that a universal logical gate set can be implemented on encoded qubits through a constant depth unitary quantum circuit, and without increasing the asymptotic scaling of the space overhead. Our result reformulates the notion of anyon braiding as an effectively instantaneous process, rather than as an adiabatic, slow process.
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.
A fundamental question in the theory of quantum computation is to understand the ultimate space-time resource costs for performing a universal set of logical quantum gates to arbitrary precision. Here we demonstrate that non-Abelian anyons in Turaev-Viro quantum error correcting codes can be moved over a distance of order of the code distance, and thus braided, by a constant depth local unitary quantum circuit followed by a permutation of qubits. Our gates are protected in the sense that the lengths of error strings do not grow by more than a constant factor. When applied to the Fibonacci code, our results demonstrate that a universal logical gate set can be implemented on encoded qubits through a constant depth unitary quantum circuit, and without increasing the asymptotic scaling of the space overhead. These results also apply directly to braiding of topological defects in surface codes. Our results reformulate the notion of braiding in general as an effectively instantaneous process, rather than as an adiabatic, slow process.
Low-depth random circuit codes possess many desirable properties for quantum error correction but have so far only been analyzed in the code capacity setting where it is assumed that encoding gates and syndrome measurements are noiseless. In this work, we design a fault-tolerant distillation protocol for preparing encoded states of one-dimensional random circuit codes even when all gates and measurements are subject to noise. This is sufficient for fault-tolerant quantum memory since these encoded states can then be used as ancillas for Steane error correction. We show through numerical simulations that our protocol can correct erasure errors up to an error rate of 2 % . In addition, we also extend results in the code capacity setting by developing a maximum likelihood marginal decoder for depolarizing noise similar to work by Darmawan []. As in their work, we formulate the decoding problem as a tensor network contraction and show how to contract the network efficiently by exploiting the low-depth structure. Replacing the tensor network with a so-called “tropical” tensor network, we also show how to perform minimum weight decoding. With these decoders, we are able to numerically estimate the depolarizing error threshold of finite-rate random circuit codes and show that this threshold closely matches the hashing bound even when the decoding is suboptimal.