On quantum tensor product codes

We present a general framework for the construction of quantum tensor product codes (QTPC). In a classical tensor product code (TPC), its parity check matrix is con- structed via the tensor product of parity check matrices of the two component codes. We show that by adding some constraints on the component codes, several classes of dual-containing TPCs can be obtained. By selecting different types of component codes, the proposed method enables the construction of a large family of QTPCs and they can provide a wide variety of quantum error control abilities. In particular, if one of the component codes is selected as a burst-error-correction code, then QTPCs have quantum multiple-burst-error-correction abilities, provided these bursts fall in distinct subblocks. Compared with concatenated quantum codes (CQC), the component code selections of QTPCs are much more exible than those of CQCs since only one of the component codes of QTPCs needs to satisfy the dual-containing restriction. We show that it is possible to construct QTPCs with parameters better than other classes of quantum error-correction codes (QECC), e.g., CQCs and quantum BCH codes. Many QTPCs are obtained with parameters better than previously known quantum codes available in the literature. Several classes of QTPCs that can correct multiple quantum bursts of errors are constructed based on reversible cyclic codes and maximum-distance-separable (MDS) codes.

[1]  J.K. Wolf,et al.  An Introduction to Tensor Product Codes and Applications to Digital Storage Systems , 2006, 2006 IEEE Information Theory Workshop - ITW '06 Chengdu.

[2]  Farrokh Vatan,et al.  Spatially correlated qubit errors and burst-correcting quantum codes , 1997, IEEE Trans. Inf. Theory.

[3]  Eric M. Rains.  Quantum Codes of Minimum Distance Two , 1999, IEEE Trans. Inf. Theory.

[4]  Mustafa Nazmi Kaynak,et al.  Classification Codes for Soft Information Generation from Hard Flash Reads , 2014, IEEE Journal on Selected Areas in Communications.

[5]  Shu Lin,et al.  Error control coding : fundamentals and applications , 1983 .

[6]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[8]  Jack K. Wolf,et al.  On codes derivable from the tensor product of check matrices , 1965, IEEE Trans. Inf. Theory.

[9]  Hideki Imai,et al.  Generalized tensor product codes , 1981, IEEE Trans. Inf. Theory.

[10]  Santosh Kumar,et al.  Nonbinary Stabilizer Codes Over Finite Fields , 2005, IEEE Transactions on Information Theory.

[11]  Matthias Schroder.  Self Dual Codes And Invariant Theory , 2016 .

[12]  Shu Lin,et al.  Applied Algebra, Algebraic Algorithms and Error-Correcting Codes , 1999, Lecture Notes in Computer Science.

[13]  P. Chaichanavong,et al.  Tensor-product Parity codes: combination with constrained codes and application to perpendicular recording , 2006, IEEE Transactions on Magnetics.

[14]  I. Devetak,et al.  General entanglement-assisted quantum error-correcting codes , 2007, 2007 IEEE International Symposium on Information Theory.

[15]  N. J. A. Sloane,et al.  Quantum Error Correction Via Codes Over GF(4) , 1998, IEEE Trans. Inf. Theory.

[16]  Martin Bossert,et al.  On the equivalence of generalized concatenated codes and generalized error location codes , 2000, IEEE Trans. Inf. Theory.

[17]  Igor Devetak,et al.  Correcting Quantum Errors with Entanglement , 2006, Science.

[18]  Jaekyun Moon,et al.  An Iteratively Decodable Tensor Product Code with Application to Data Storage , 2010, IEEE Journal on Selected Areas in Communications.

[19]  D. Vernon.  Inform , 1995, Encyclopedia of the UN Sustainable Development Goals.

[20]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[21]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[22]  P. Chaichanavong,et al.  Tensor-product parity code for magnetic recording , 2006, IEEE Transactions on Magnetics.

[23]  Hanwu Chen,et al.  A Construction of Quantum Error-Locating Codes , 2017 .

[24]  Li-Yi Hsu,et al.  High Performance Entanglement-Assisted Quantum LDPC Codes Need Little Entanglement , 2009, IEEE Transactions on Information Theory.

[25]  Martin Rötteler,et al.  Quantum block and convolutional codes from self-orthogonal product codes , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[26]  V. Giovannetti,et al.  Quantum channels and memory effects , 2012, 1207.5435.

[27]  Martin Rötteler,et al.  On quantum MDS codes , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[28]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[29]  Raymond Laflamme,et al.  Concatenated Quantum Codes , 1996 .

[30]  Paul H. Siegel,et al.  Binary Linear Locally Repairable Codes , 2015, IEEE Transactions on Information Theory.

[31]  Lara Dolecek,et al.  Graded Bit-Error-Correcting Codes With Applications to Flash Memory , 2013, IEEE Transactions on Information Theory.

[32]  Giuliano G. La Guardia,et al.  ASYMMETRIC QUANTUM PRODUCT CODES , 2012 .

[33]  Markus Grassl,et al.  Generalized concatenated quantum codes , 2009 .

[34]  W. Bosma,et al.  HANDBOOK OF MAGMA FUNCTIONS , 2011 .

[35]  Xueliang Li,et al.  Binary construction of quantum codes of minimum distances five and six , 2008, Discret. Math..

[36]  J. A. S. Kelso,et al.  Information and control , 1987 .

[37]  Chaoping Xing,et al.  Generalization of Steane's Enlargement Construction of Quantum Codes and Applications , 2010, IEEE Transactions on Information Theory.

[38]  S. Kawabata.  Quantum Interleaver. Quantum Error Correction for Burst Error. , 2000, quant-ph/0002020.

[39]  Helmut Grießer.  On soft concatenated decoding of block codes , 2003 .

[40]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[41]  Pradeep Kiran Sarvepalli,et al.  On Quantum and Classical BCH Codes , 2006, IEEE Transactions on Information Theory.

[42]  Joseph L. Yucas,et al.  Self-Reciprocal Irreducible Polynomials Over Finite Fields , 2004, Des. Codes Cryptogr..

[43]  Markus Grassl,et al.  Quantum Reed-Solomon Codes , 1999, AAECC.

[44]  M. Darnell,et al.  Error Control Coding: Fundamentals and Applications , 1985 .

[45]  Martin Bossert,et al.  Some Results on Generalized Concatenation of Block Codes , 1999, AAECC.

[46]  Hanwu Chen,et al.  Comments on and Corrections to “On the Equivalence of Generalized Concatenated Codes and Generalized Error Location Codes” , 2017, IEEE Transactions on Information Theory.

[47]  Steane,et al.  Error Correcting Codes in Quantum Theory. , 1996, Physical review letters.

[48]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

[49]  James L. Massey.  Reversible Codes , 1964, Inf. Control..

[50]  C. Mitchell.  in Designs , Codes , and Cryptography , 2007 .