Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically

Fan, J; Li, J; Wang, Y; Li, Y; Hsieh, MH; Du, J

Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically

Fan, J Li, J Wang, Y Li, Y Hsieh, MH Du, J

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Publisher:: Institute of Electrical and Electronics Engineers (IEEE)
Publication Type:: Journal Article
Citation:: IEEE Transactions on Information Theory, 2023, 69, (1), pp. 262-272
Issue Date:: 2023-01-01

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Field	Value	Language
dc.contributor.author	Fan, J
dc.contributor.author	Li, J
dc.contributor.author	Wang, Y
dc.contributor.author	Li, Y
dc.contributor.author	Hsieh, MH
dc.contributor.author	Du, J
dc.date.accessioned	2023-03-14T04:51:29Z
dc.date.available	2023-03-14T04:51:29Z
dc.date.issued	2023-01-01
dc.identifier.citation	IEEE Transactions on Information Theory, 2023, 69, (1), pp. 262-272
dc.identifier.issn	0018-9448
dc.identifier.issn	1557-9654
dc.identifier.uri	http://hdl.handle.net/10453/167303
dc.description.abstract	In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor- Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane's enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parameters Q1 = [[N,Ω( √ N), Ω( √ N)]], and Q2 = [[N, Ω(N/ logN), Ω(N/ logN)/Ω(logN)]]. We show that Q1 can be encoded very efficiently by circuits of size O(N) and depth O( √ N). For an input error syndrome, Q1 can correct any adversarial error of weight up to half the minimum distance bound in O(N) time. Q1 can also be decoded in parallel in O( √ N) time by using O( √ N) classical processors. For an input error syndrome, we proved that Q2 can correct a linear number of X-errors with high probability and an almost linear number of Z-errors in O(N) time. Moreover, Q2 can be decoded in parallel in O(log(N)) time by using O(N) classical processors.
dc.language	en
dc.publisher	Institute of Electrical and Electronics Engineers (IEEE)
dc.relation.ispartof	IEEE Transactions on Information Theory
dc.relation.isbasedon	10.1109/TIT.2022.3201239
dc.rights	info:eu-repo/semantics/closedAccess
dc.subject	0801 Artificial Intelligence and Image Processing, 0906 Electrical and Electronic Engineering, 1005 Communications Technologies
dc.subject.classification	Networking & Telecommunications
dc.title	Partially Concatenated Calderbank-Shor-Steane Codes Achieving the Quantum Gilbert-Varshamov Bound Asymptotically
dc.type	Journal Article
utslib.citation.volume	69
utslib.for	0801 Artificial Intelligence and Image Processing
utslib.for	0906 Electrical and Electronic Engineering
utslib.for	1005 Communications Technologies
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Strength - QSI - Centre for Quantum Software and Information
utslib.copyright.status	closed_access	*
dc.date.updated	2023-03-14T04:51:28Z
pubs.issue	1
pubs.publication-status	Published
pubs.volume	69
utslib.citation.issue	1

Abstract:

In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor- Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane's enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parameters Q1 = [[N,Ω( √ N), Ω( √ N)]], and Q2 = [[N, Ω(N/ logN), Ω(N/ logN)/Ω(logN)]]. We show that Q1 can be encoded very efficiently by circuits of size O(N) and depth O( √ N). For an input error syndrome, Q1 can correct any adversarial error of weight up to half the minimum distance bound in O(N) time. Q1 can also be decoded in parallel in O( √ N) time by using O( √ N) classical processors. For an input error syndrome, we proved that Q2 can correct a linear number of X-errors with high probability and an almost linear number of Z-errors in O(N) time. Moreover, Q2 can be decoded in parallel in O(log(N)) time by using O(N) classical processors.

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http://hdl.handle.net/10453/167303