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New quantum codes constructed from quaternary BCH codes

quantum INFORMATION PROCESSING 2016年第10期15卷 4099-4116页

作者： Xu, Gen Li, Ruihu Guo, Luobin Ma, Yuena Air Force Engn Univ Dept Math Coll Sci Xian 710051 Shaanxi Peoples R China

In this paper, we firstly study construction of new quantum error-correcting codes (QECCs) from three classes of quaternary imprimitive BCH codes. As a result, the improved maximal designed distance of these narrow-sense imprimitive Hermitian dual-containing quaternary BCH codes are determined to be much larger than the result given according to Aly et al. (IEEE Trans Inf Theory 53: 1183-1188, 2007) for each different code length. Thus, families of new QECCs are newly obtained, and the constructed QECCs have larger distance than those in the previous literature. Secondly, we apply a combinatorial construction to the imprimitive BCH codes with their corresponding primitive counterpart and construct many new linear quantum codes with good parameters, some of which have parameters exceeding the finite Gilbert-Varshamov bound for linear quantum codes.

关键词： BCH code Maximal designed distance quantum error-correcting code Combinatorial construction Finite Gilbert-Varshamov bound

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IEEE TRANSACTIONS ON INFORMATION THEORY 1999年第5期45卷 1703-1708页

作者： Vatan, F Roychowdhury, VP Anantram, MP Univ Calif Los Angeles Dept Elect Engn Los Angeles CA 90095 USA NASA Ames Res Ctr Moffett Field CA 94035 USA

We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of burst errors in classical coding theory. We present several different efficient schemes for constructing families of such codes. For example, one can find one-dimensional quantum codes of length n = 13 and 15 that correct burst errors of width b less than or equal to 3;as a comparison, a random-error correcting quantum code that corrects t = 3 errors must have length n greater than or equal to 17. In general, we show that it is possible to build quantum burst-correcting codes that have near optimal dimension. For example, we show that for any constant b, there exist b-burst-correcting quantum codes with length n, and dimension k = n - log n - O(1);as a comparison, the Hamming bound for the case with t (constant) random errors yields k less than or equal to n - t log n + O(1).

关键词： additive codes burst errors cyclic codes Gilbert-Varshamov bound quantum error-correcting code

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Message randomization and strong security in quantum stabilizer-based secret sharing for classical secrets

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DESIGNS codeS AND CRYPTOGRAPHY 2020年第9期88卷 1893-1907页

作者： Matsumoto, Ryutaroh Tokyo Inst Technol Dept Informat & Commun Engn Tokyo Japan Aalborg Univ Dept Math Sci Aalborg Denmark

We improve the flexibility in designing access structures of quantum stabilizer-based secret sharing schemes for classical secrets, by introducing message randomization in their encoding procedures. We generalize the Gilbert-Varshamov bound for deterministic encoding to randomized encoding of classical secrets. We also provide an explicit example of a ramp secret sharing scheme with which multiple symbols in its classical secret are revealed to an intermediate set, and justify the necessity of incorporating strong security criterion of conventional secret sharing. Finally, we propose an explicit construction of strongly secure ramp secret sharing scheme by quantum stabilizers, which can support twice as large classical secrets as the McEliece-Sarwate strongly secure ramp secret sharing scheme of the same share size and the access structure.

关键词： Secret sharing quantum error-correcting code Gilbert-Varshamove bound Strong security

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A new approach to constructing CSS codes based on factor graphs

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INFORMATION SCIENCES 2008年第7期178卷 1893-1902页

作者： Wang, Huisong Wang, Jun Du, Qun Zeng, Guihua Shanghai Jiao Tong Univ Dept Elect Engn State Key Lab FiberOpt Local Area Networks & Adv Shanghai 200030 Peoples R China Cent Architectural Design Inst Wuhan 430071 Peoples R China

A novel, practical and convenient approach to constructing Calderbank-Shor Steane (CSS) codes based on factor graphs is presented in this paper. Our proposed method is applied to solve two problems associated with constructing CCS codes. One is judging whether a code is a weakly self-dual code or not, the other is finding the generator matrix and parity-check matrix of a weakly self-dual code. The novelty, practicality and convenience of the approach are shown as follows. First, the approach is a hitherto unexplored one to constructing CSS codes. Second, the judgment of a weakly self-dual code is entirely based on factor graphs. Namely, we consider a code a weakly self-dual one when the Tanner graph or convolutional factor graph of its dual code can be obtained by that of its own via our proposed transform TR-->L. Finally, we can obtain the generator matrix and parity-check Matrix of a weakly self-dual code via factor graphs other than conventional algebra methods, which allow us avoid matrix Computation to get them. An example is given to show how to construct quantum CSS code based on factor graphs. The method can be extended to other CSS codes. (C) 2007 Elsevier Inc. All rights reserved.

关键词： quantum error-correcting code CSS code factor graph duality

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A Construction of quantum error-Locating codes

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Communications in Theoretical Physics 2017年第1期67卷 37-40页

作者：樊继豪陈汉武 School of Computer Science and Engineering Southeast UniversityNanjing 211189China Key Laboratory of Computer Network and Information Integration(Southeast University) Ministry of EducationNanjing 211189China

We present the construction of quantum error-locating(QEL) codes based on classical error-locating(EL)codes. Similar to classical EL codes, QEL codes lie midway between quantum error-correcting codes and quantum errordetecting codes. Then QEL codes can locate qubit errors within one sub-block of the received qubit symbols but do not need to determine the exact locations of the erroneous qubits. We show that, an e-error-locating code derived from an arbitrary binary cyclic code with generator polynomial g(x), can lead to a QEL code with e error-locating abilities, only if g(x) does not contain the(1 + x)-factor.

关键词： quantum error-correcting code error-locating code cyclic code

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Application of GRS codes to some entanglement-assisted quantum MDS codes

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quantum INFORMATION PROCESSING 2022年第3期21卷 1-16页

作者： Wang, Guohui Tang, Chunming Guangzhou Univ Sch Math & Informat Sci Guangzhou 510006 Peoples R China Guangdong Prov Key Lab Informat Secur Technol Guangzhou 510006 Peoples R China Qinghai Univ Basic Course Teaching & Res Dept Xining 810016 Peoples R China

Entanglement-assisted quantum error-correcting codes (EAQECCs) are a generalization of standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing pre-shared entanglement between the sender and receiver to boost the rate of transmission. In this paper, we present some constructions of generalized Reed-Solomon codes and calculate the dimension of their hulls. With these generalized Reed-Solomon codes, we obtain four infinite families new constructions of EAQECCs, which are optimal with respect to the Singleton bound for EAQECCs. Notably, the parameters of our EAQECCs are flexible.

关键词： Entanglement-assisted quantum MDS code quantum error-correcting code Generalized Reed-Solomon code

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Structure of CSS and CSS-T quantum codes

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DESIGNS codeS AND CRYPTOGRAPHY 2024年第10期92卷 2801-2823页

作者： Berardini, Elena Caminata, Alessio Ravagnani, Alberto CNRS 351 Cours Liberat F-33405 Talence France Univ Bordeaux IMB 351 Cours Liberat F-33405 Talence France Univ Genoa Dipartimento Matemat Dipartimento Eccellenza 2023 2027 Via Dodecaneso 35 I-16146 Genoa Italy Eindhoven Univ Technol Eindhoven Netherlands

We investigate CSS and CSS-T quantum error-correcting codes from the point of view of their existence, rarity, and performance. We give a lower bound on the number of pairs of linear codes that give rise to a CSS code with good correction capability, showing that such pairs are easy to produce with a randomized construction. We then prove that CSS-T codes exhibit the opposite behaviour, showing also that, under very natural assumptions, their rate and relative distance cannot be simultaneously large. This partially answers an open question on the feasible parameters of CSS-T codes. We conclude with a simple construction of CSS-T codes from Hermitian curves. The paper also offers a concise introduction to CSS and CSS-T codes from the point of view of classical coding theory.

关键词： quantum error-correcting code CSS code CSS-T code code parameters

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Gapped boundaries, group cohomology and fault-tolerant logical gates

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ANNALS OF PHYSICS 2017年 377卷 387-413页

作者： Yoshida, Beni CALTECH Walter Burke Inst Theoret Phys Pasadena CA 91125 USA Perimeter Inst Theoret Phys Waterloo ON N2L 2Y5 Canada Univ Calif Santa Barbara Kavli Inst Theoret Phys Santa Barbara CA 93106 USA

This paper attempts to establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error correcting codes. We begin by presenting constructions of gapped boundaries for the d-dimensional quantum double model by using d-cocycles functions (d >= 2). We point out that the system supports m-dimensional excitations (m < d), which we shall call fluctuating charges, that are superpositions of point-like electric charges characterized by m-dimensional bosonic SPT wavefunctions. There exist gapped boundaries where electric charges or magnetic fluxes may not condense by themselves, but may condense only when accompanied by fluctuating charges. Magnetic fluxes and codimension-2 fluctuating charges exhibit non-trivial multi-excitation braiding statistics, involving more than two excitations. The statistical angle can be computed by taking slant products of underlying cocycle functions sequentially. We find that excitations that may condense into a gapped boundary can be characterized by trivial multi-excitation braiding statistics, generalizing the notion of the Lagrangian subgroup. As an application, we construct fault-tolerantly implementable logical gates for the d-dimensional quantum double model by using d-cocycle functions. Namely, corresponding logical gates belong to the dth level of the Clifford hierarchy, but are outside of the (d - 1)th level, if cocycle functions have non-trivial sequences of slant products. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Gapped boundary quantum error-correcting code Topological order Fault-tolerance

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Constructing quantum codes

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中国高等学校学术文摘·计算机科学 2008年第2期2卷 143-146页

作者： Hao CHEN Software Engineering Institute East China Normal UniversityShanghai 200062China

quantum error correcting codes are indispensable for quantum information processing and quantum *** 1995 and 1996,Shor and Steane gave first several examples of quantum codes from classical error correcting *** construction of efficient quantum codes is now an active multi-discipline research *** this paper we review the known several constructions of quantum codes and present some examples.

关键词： quantum error-correcting code CSS construction enlargement of the CSS code symplectic inner product twisted code AG-code

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A quantum Stabilizer code Associated with Cluster States

A Quantum Stabilizer Code Associated with Cluster States

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International Conference on the Frontiers and Advances in Data Science (FADS)

作者： Song, Dan Cao, Zhengwen Zhang, Shuanghao Feng, Jie Li, Yan Chai, Geng Northwest Univ Sch Informat Sci & Technol Xian Shaanxi Peoples R China

ISBN: (纸本)9781538631485

We present a construction scheme for quantum error-correcting code of cluster states, which is a similar method of splicing stepwise as the Toric code. The stabilizer codes space can be founded by a Hamiltonian, which is constructed with a finite abelian group. A principal code space is set up by the entanglement among cluster state particles. Then a complete stabilizer code matrix can be found by supplement of computing analysis which satisfies the error-correcting conditions. This allows a simple verification for 1 - error correcting property in any dimension. Furthermore, we can obtain a suitable quantum stabilizer code.

关键词： quantum error-correcting code quantum stabilizer code Cluster states Toric code Topology construction

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