Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes

Sabo, Eric; Aloshious, Arun B.; Brown, Kenneth R.

doi:10.48550/arxiv.2106.08251

Cited by 4 publications

(6 citation statements)

References 39 publications

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“…This causes stabilizer codes to exhibit a particular coset structure in which multiple different error patterns act identically on the transmitted information [58,65,66]. Although the manifestation of degeneracy in the design of sparse quantum codes and its effects on the decoding process has been studied extensively [16,55,56,67,68,69,70,71], especially for QTCs and quantum topological codes [29,68,72,73,74], it remains a somewhat obscure topic in the literature. This can be attributed to the varying and sometimes inconsistent notation and the oft confusing nature of the notion of degeneracy itself.…”

Section: Understanding and Exploiting Degeneracymentioning

confidence: 99%

“…Furthermore, we will restrict the discussion to the sum-product based decoding of stabilizer codes, which (mostly) concerns QLDPC and QTC codes [46,110,111,112]. Certain families of stabilizer codes, such as topological codes [72,73,74], can be decoded differently and so this discussion may not apply to them.…”

Section: Decoding Quantum Stabilizer Codesmentioning

confidence: 99%

“…Fortunately, a metric capable of telling different end-to-end errors apart already exists. It is known as the logical error rate and it has been widely employed to assess the performance of various other families of QEC codes such as QTCs and quantum topological codes [68,29,72,73,74]. Generally, the decoders used in QTC or topological schemes are capable of producing estimates of the stabilizer coset representatives (the logical operators) of the channel errors, hence these decoders "inherently" compute the logical error rate.…”

Section: Physical and Logical Error Ratementioning

confidence: 99%

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Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes

et al. 2020

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The game took me to Saint Andrew's College and so we now turn to that marvelous place. I would not be who I am today had I not spent two years at SAC. True to its motto, the place molded me into a significantly more mature individual and taught me skills that have shined with brilliance during my time as a PhD student. To all my teachers, coaches, and friends at SAC, thank you. Special shoutouts to my friends in 1st Hockey and the so-called "Dawgz": Graham, Humza, David, YoungWoo, West, and Andy. Although we have not seen each other in a long time, I know our friendship still runs as deep as it did when we gamed away our days at the Manor. Following this theme of childhood friends, thank you to Isma, Guille, Luken, and Andrey. From the day we first met in kindergarten at Saint Patrick's, our passion for sports and later on fitness kept us united. I pray that our games of pádel never stop being so competitive.Barring chance or extremely good fortune, those I will thank next will likely never read this dissertation. Thank you to R. A Salvatore, Steven Erikson, Brandon Sanderson, Patrick Rothfuss, and all those other authors from whom I have learned so much. Thank you also to Satoshi Nakamoto. I do not doubt that your creation will change the world for the better. I must also thank the Counting Crows, Rise Against, Machine Gun Kelly, and the Chainsmokers for motivating me in all those instances when my discipline evaporated. In relation to this, I must also thank the people at Youtube and Nespresso (and Iñigo Gutierrez by association); procrastination via the classic video and black coffee combo will never get old. Finally, a special shoutout to all those online meetings, home workouts, and quarantine periods brought to me by the COVID-19 pandemic. Somehow, throughout this entire ordeal, my work has not been impeded at any point in time. I have been extremely fortunate.Having left the best for last, I must now turn to those closest to my heart. After spending days thinking about how to thank my parents appropriately, I found it best to use the following quote by the writer Chuck Palahniuk: "First your parents, they give you your life, but then they try to give you their life". Despite this, I am still left with a feeling of insufficiency. I guess my parents have done so much for me that it is not possible to put my gratitude into words. Gracias attatto y amatxo por todo. The same goes for my sister Idoia, to whom I owe more than can be expressed. Thank you for always being there, for always having time to talk, and most of all, for always being willing to listen. Through thick and thin, your presence has never failed to remind me that there is always light at the end of the tunnel. Lastly, thank you Moose for being such an energetic furball and for greeting me at the door everyday as if you had not seen me in years.Alas, it is now time for me to thank one final person. Maria, you, above anyone else, have lived through the highs and the lows of my PhD. You have seen the extent to which manuscript revisions can frustr...

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Section: Understanding and Exploiting Degeneracymentioning

confidence: 99%

Section: Decoding Quantum Stabilizer Codesmentioning

confidence: 99%

Section: Physical and Logical Error Ratementioning

confidence: 99%

See 1 more Smart Citation

Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes

et al. 2020

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“…MWPM achieves a threshold of 13.3% on the (4,8,8) color codes without specifying the complexity [30] and a threshold of 13.05% on the (6,6,6) color codes with complexity O(N 4 ) [31]. (Decoding the (4,8,8) color codes is considered relatively harder from the trellis complexity of the code [34].) AMBP 4 , on the other hand, can decode a color code by just giving its check matrix, without additional processes.…”

Section: Decoding Performancementioning

confidence: 99%

Comparison of 2D topological codes and their decoding performances

Kuo¹,

Lai²

2022

Preprint

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Topological quantum codes are favored because they allow qubit layouts that are suitable for practical implementation. An N -qubit topological code can be decoded by minimum-weight perfect matching (MWPM) with complexity O(poly(N )) if it is of CSS-type. Recently it is shown that various quantum codes, including non-CSS codes, can be decoded by an adapted belief propagation with memory effects (denoted MBP) with complexity almost linear in N . In this paper, we show that various twodimensional topological codes, CSS or non-CSS, regardless of the layout, can be decoded by MBP, including color codes and twisted XZZX codes. We will comprehensively compare these codes in terms of code efficiency and decoding performance, assuming perfect error syndromes.

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“…The logical error rate is a well-known metric and has been widely employed to assess the performance of other families of Quantum Error Correction (QEC) codes such as Quantum Turbo Codes (QTC) and quantum topological codes [10], [35]- [38]. For these particular error correction VOLUME 4, 2016 schemes, the decoders that are employed are sometimes capable of distinguishing between error equivalence classes (they can account for some aspects of degeneracy), which makes it easy to compute the logical error rate.…”

Section: Introductionmentioning

confidence: 99%

On the Logical Error Rate of Sparse Quantum Codes

Fuentes

Martinez

Crespo

et al. 2022

IEEE Trans. Quantum Eng.

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The quantum paradigm presents a phenomenon known as degeneracy that can potentially improve the performance of quantum error correcting codes. However, the effects of this mechanism are sometimes ignored when evaluating the performance of sparse quantum codes and the logical error rate is not always correctly reported. In this paper, we discuss previously existing methods to compute the logical error rate and we present an efficient coset-based method inspired by classical coding strategies to estimate degenerate errors and distinguish them from logical errors. Additionally, we show that the proposed method presents a computational advantage for the family of Calderbank-Shor-Steane (CSS) codes. We use this method to prove that degenerate errors are frequent in a specific family of sparse quantum codes, which stresses the importance of accurately reporting their performance. Our results also reveal that the modified decoding strategies proposed in the literature are an important tool to improve the performance of sparse quantum codes.INDEX TERMS Quantum error correction, quantum low density parity check codes, quantum low density generator matrix codes, iterative decoding.

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Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes

Cited by 4 publications

References 39 publications

Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes

Approach for the construction of non-Calderbank-Steane-Shor low-density-generator-matrix–based quantum codes

Comparison of 2D topological codes and their decoding performances

On the Logical Error Rate of Sparse Quantum Codes

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