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arXiv:1302.3638 · Physical Review A

Kitaev's Zd -code threshold estimates

Guillaume Duclos-Cianci, David Poulin

Published Jun 27, 2013• 51 citations
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Abstract

We study the quantum error correction threshold of Kitaev's toric code over the group ${\mathbb{Z}}_{d}$ subject to a generalized bit-flip noise. This problem requires special decoding techniques, and for this purpose we generalize the renormalization-group method we introduced previously [G. Duclos-Cianci and D. Poulin, Phys. Rev. Lett. 104, 050504 (2010) and IEEE Information Theory Workshop, Dublin (2010), p. 1] for ${\mathbb{Z}}_{2}$ topological codes.

DOI
10.1103/physreva.87.062338

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We study the quantum error correction threshold of Kitaev's toric code over the group Zd{\mathbb{Z}}_{d}Zd​ subject to a generalized bit-flip noise. This problem requires special decoding techniques, and for this purpose we generalize the renormalization-group method we introduced previously [G. Duclos-Cianci and D. Poulin, Phys. Rev. Lett. 104, 050504 (2010) and IEEE Information Theory Workshop, Dublin (2010), p. 1] for Z2{\mathbb{Z}}_{2}Z2​ topological codes.
Qudit toric codes are a natural higher-dimensional generalization of the well-studied qubit toric code. However standard methods for error correction of the qubit toric code are not applicable to them. Novel decoders are needed. In this paper we introduce two renormalization group decoders for qudit codes and analyze their error correction thresholds and efficiency. The first decoder is a generalization of a "hard-decisions" decoder due to Bravyi and Haah [arXiv:1112.3252]. We modify this decoder to overcome a percolation effect which limits its threshold performance for high dimensions. The second decoder is a generalization of a "soft-decisions" decoder due to Poulin and Duclos-Cianci [Phys. Rev. Lett. 104, 050504 (2010)], with a small cell size to optimize the efficiency of implementation in the high dimensional case. In each case, we estimate thresholds for the uncorrelated bit-flip error model and provide a comparative analysis of the performance of both these approaches to error correction of qudit toric codes.
We study the error thresholds of a new topological quantum error-correcting code under the independent bit-flip and phase-flip noisy channel. By mapping to classical spin-glass models, we find that if Pauli X and Z errors are equally probable, the threshold of vertex syndromes is improved by 30% compared to that of the toric code, while the threshold of plaquette syndromes is lowered. Both thresholds increase when the Z component is less likely than the X component, reaching 0.5 if the Z component is absent. An analytical proof justifies this special case of perfect correctability, and a disordered version of Peierls's argument establishes a lower bound for the vertex-syndrome threshold.
Kitaev's toric code is arguably the most studied quantum code and is expected to be implemented in future generations of quantum computers. The renormalisation decoders introduced by Duclos-Cianci and Poulin exhibit one of the best trade-offs between efficiency and speed, but one question that was left open is how they handle worst-case or adversarial errors, i.e. what is the order of magnitude of the smallest weight of an error pattern that will be wrongly decoded. We initiate such a study involving a simple hard-decision and deterministic version of a renormalisation decoder. We exhibit an uncorrectable error pattern whose weight scales like d1/2and prove that the decoder corrects all error patterns of weight less than 56dlog⁡2(6/5)\frac{5}{6}{d^{{{\log }_2}(6/5)}}65​dlog2​(6/5), where d is the minimum distance of the toric code.
Topological quantum error correction codes are currently among the most promising candidates for efficiently dealing with the decoherence effects inherently present in quantum devices. Numerically, their theoretical error threshold can be calculated by mapping the underlying quantum problem to a related classical statistical-mechanical spin system with quenched disorder. Here, we present results for the general fault-tolerant regime, where we consider both qubit and measurement errors. However, unlike in previous studies, here we vary the strength of the different error sources independently. Our results highlight peculiar differences between toric and color codes. This study complements previous results published in New J. Phys. 13, 083006 (2011).
We compute the error threshold for the semion code, the companion of the Kitaev toric code with the same gauge symmetry group Z2{\mathbb{Z}}_{2}Z2​. The application of statistical mechanical mapping methods is highly discouraged for the semion code, since the code is non-Pauli and non-Calderbank-Shor-Steane (CSS). Thus, we use machine learning methods, taking advantage of the near-optimal performance of some neural network decoders: multilayer perceptrons and convolutional neural networks (CNNs). We find the values peff=9.5{p}_{\text{eff}}=9.5%peff​=9.5 for uncorrelated bit-flip and phase-flip noise, and peff=10.5{p}_{\text{eff}}=10.5%peff​=10.5 for depolarizing noise. We contrast these values with a similar analysis of the Kitaev toric code on a hexagonal lattice with the same methods. For convolutional neural networks, we use the ResNet architecture, which allows us to implement very deep networks and results in better performance and scalability than the multilayer perceptron approach. We analyze and compare in detail both approaches and provide a clear argument favoring the CNN as the best suited numerical method for the semion code.