A renormalization group decoding algorithm for topological quantum codes

Duclos-Cianci, Guillaume; Poulin, David

doi:10.1109/cig.2010.5592866

Cited by 50 publications

(65 citation statements)

References 12 publications

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“…Nevertheless, the main disadvantage of the MWPMA is that it is not suitable for qudit surface codes with d > 2. For these reasons, the development of alternative decoding algorithms is currently a very active research area [14,15,[22][23][24][25][26][27].…”

Section: Overviewmentioning

confidence: 99%

“…5, where we find a threshold value of p (2) th = 0.0215 ± 0.0006. This allows us to directly compare our decoder with other fault-tolerant qubit decoders; for example, the soft-decision renormalization-group decoder by Duclos-Cianci and Poulin achieves a threshold of p (2) th = 0.019 ± 0.004 [22].…”

Section: Thresholds Estimation and Percolation Limitationmentioning

confidence: 99%

“…This is in stark contrast to other known qudit decoders. For example, the soft-decision renormalization-group decoder in the faulttolerant setting [22] has a straightforward implementation in higher dimensions but comes with a cost of a polynomial overhead in d which means its applicability is limited to low dimensions. …”

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confidence: 99%

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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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Section: Overviewmentioning

confidence: 99%

Section: Thresholds Estimation and Percolation Limitationmentioning

confidence: 99%

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Fast fault-tolerant decoder for qubit and qudit surface codes

2015

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“…This additional complication in the structure of the syndromes makes the problem of finding optimal decoders for color codes more difficult. There are decoders based on the renormalization group [27], and decoders which decompose the decoding problem into multiple instances of minimum-weight perfect matching [19]. Using a non-scalable decoder based on integer programming, thresholds of 10.56% (data-only errors), 3.05% (data & syndrome errors) and 0.082% (circuit-based errors) have been obtained for the color code on the square-octagon lattice [16].…”

Section: Decodingmentioning

confidence: 99%

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Criger

Terhal

2016

QIC

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We analyze the properties of a 2D topological code derived by concatenating the 4, 2, 2 code with the toric/surface code, or alternatively by removing check operators from the 2D square-octagon or 4.8.8 color code. We show that the resulting code has a circuit-based noise threshold of ∼ 0.41% (compared to ∼ 0.6% for the toric code in a similar scenario), which is higher than any known 2D color code. We believe that the construction may be of interest for hardware in which one wants to use both long-range two-qubit gates as well as short-range gates between small clusters of qubits.

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“…However, in the case of surface codes, it has been shown that some proposed renormalization-group (RG) algorithms can be generalized to qudit codes in a straightforward manner [8,51]. Moreover, recent work for the 3D fault-tolerant implementation of the qudit surface code using a hard-decision RG decoder [9] suggests that adaptations of such algorithms for higher spatial dimensions should be possible, however, the error correction thresholds would likely be degraded as the spatial dimension increases due to the larger stabilizer generators.…”

Section: Error Detectionmentioning

confidence: 99%

Qudit color codes and gauge color codes in all spatial dimensions

et al. 2015

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Two-level quantum systems, qubits, are not the only basis for quantum computation. Advantages exist in using qudits, d-level quantum systems, as the basic carrier of quantum information. We show that color codes, a class of topological quantum codes with remarkable transversality properties, can be generalized to the qudit paradigm. In recent developments it was found that in three spatial dimensions a qubit color code can support a transversal non-Clifford gate and that in higher spatial dimensions additional non-Clifford gates can be found, saturating Bravyi and König's bound [S. Bravyi and R. König, Phys. Rev. Lett. 111, 170502 (2013)]. Furthermore, by using gauge fixing techniques, an effective set of Clifford gates can be achieved, removing the need for state distillation. We show that the qudit color code can support the qudit analogs of these gates and also show that in higher spatial dimensions a color code can support a phase gate from higher levels of the Clifford hierarchy that can be proven to saturate Bravyi and König's bound in all but a finite number of special cases. The methodology used is a generalization of Bravyi and Haah's method of triorthogonal matrices [S. Bravyi and J. Haah, Phys. Rev. A 86, 052329 (2012)], which may be of independent interest. For completeness, we show explicitly that the qudit color codes generalize to gauge color codes and share many of the favorable properties of their qubit counterparts.

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A renormalization group decoding algorithm for topological quantum codes

Cited by 50 publications

References 12 publications

Fast fault-tolerant decoder for qubit and qudit surface codes

Fast fault-tolerant decoder for qubit and qudit surface codes

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Qudit color codes and gauge color codes in all spatial dimensions

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