Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength

Tillich, Jean–Pierre; Zémor, Gilles

doi:10.1109/tit.2013.2292061

Cited by 176 publications

(174 citation statements)

References 36 publications

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“…An efficient decoding algorithm for the more exotic low density parity check (LDPC) code, the four-dimensional hyperbolic code, was introduced by Hastings with similar "greedy local matching" principles as the HDRG decoder [35]. Other LDPC codes exist for which efficient decoders have not yet been identified [36,37]. The development of computationally light fault-tolerant decoders for these codes is essential if they are to be practical.…”

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confidence: 99%

Fast fault-tolerant decoder for qubit and qudit surface codes

2015

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The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process-it is the algorithm which computes the operations needed to correct or compensate for the errors according to the measured syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes. In this work, we develop a fault-tolerant decoder for the surface code, capable of efficient operation for qubits and qudits of any dimension, generalizing the decoder first introduced by Bravyi and Haah [Phys. Rev. Lett. 111, 200501 (2013)]. We study its performance when both the physical qudits and the syndromes measurements are subject to generalized uncorrelated bit-flip noise (and the higher-dimensional equivalent). We show that, with appropriate enhancements to the decoder and a high enough qudit dimension, a threshold at an error rate of more than 8% can be achieved. I. OVERVIEWTopological quantum codes built from qubits [twodimensional (2D) quantum systems] play a central role in architectures for fault-tolerant quantum computing at the forefront of current research [1][2][3][4]. The surface code [5] and the related toric code [6,7] are prominent examples of such codes. Compared with other quantum error correcting codes, they posses the key experimental benefit of requiring only local interactions and yet, under realistic noise models, they have been shown to achieve the highest reported fault-tolerant thresholds [8,9].Recent developments have shown that employing ddimensional quantum systems, or qudits, as the building blocks for fault-tolerant schemes may offer some important advantages. For example, an integral part of many faulttolerant schemes is the distillation of magic states [10]-a procedure necessary to achieve universal computation-where generalization to higher dimensions has resulted in improved distillation thresholds and lower overheads in the number of qudit magic states [11][12][13]. Moreover, threshold investigations of the qudit toric code with noise-free syndrome measurements have shown that, for a standard independent noise model, the error correction threshold increases significantly with increasing qudit dimension [14][15][16], although we caution that it is difficult to fairly compare noise rates between systems of different dimensions. Although it is more challenging to realize qudit quantum systems experimentally, recent work has demonstrated the ability to coherently control and perform operations in single 16-dimensional atomic systems with high fidelity [17,18], with the implementation of high-fidelity multiqudit interactions still to be achieved.A surface code is a stabilizer code with local stabilizer generators. Qudits are associated with the edges of a 2D square lattice. In order to store the encoded information for * fern.watson10@imperial...

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confidence: 99%

Fast fault-tolerant decoder for qubit and qudit surface codes

2015

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“…A low overhead is a desirable property when considering large scale implementation. Given the challenge of controlling hundreds of physical qubits a scheme which can deliver more logical qubits, with similar decoding performance and system size, is desirable [20,21,22]. Several code families exist which have a finite rate.…”

Section: Overheadmentioning

confidence: 99%

“…There exist classes of LDPC codes which do not fall within the topological or convolutional code classes. These typically involve a random or pseudo random construction, as in the case of bicycle codes [84,40,85] or graph based codes [22,86,40,85,87]. …”

Section: Brief History Of Quantum Codesmentioning

confidence: 99%

“…Hyper bicycle or finite-rate distance scaling codes are an attractive possibility for large scale quantum systems [22,138]. They combine the decoding strength and distance scaling capacity of topological codes with the finite rate properties of certain LDPC codes, such as convolutional codes.…”

Section: Hyper Bicycle Codesmentioning

confidence: 99%

“…There is a great deal of interest regarding quantum LDPC codes [86,123,84,22,135,40,118]. In part this is driven by the broadness of the general description itself, requiring only that the codes have low weight stabilizers.…”

Section: Other Ldpc Codes In the Foliated Settingmentioning

confidence: 99%

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