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Nonbinary quantum Reed-Muller codes
- In Proc. Int. Symp. Inform. Theory
, 2005
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Nonbinary Stabilizer Codes
"... Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposi ..."
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Cited by 2 (2 self)
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Recently, the field of quantum error-correcting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum error-correction. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.
CODES AS FRACTALS AND NONCOMMUTATIVE SPACES
, 1107
"... Abstract. We consider the CSS algorithm relating self-orthogonal classical linear codes to q-ary quantum stabilizer codes and we show that to such a pair of a classical and a quantum code one can associate geometric spaces constructed using methods from noncommutative geometry, arising from rational ..."
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Cited by 1 (1 self)
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Abstract. We consider the CSS algorithm relating self-orthogonal classical linear codes to q-ary quantum stabilizer codes and we show that to such a pair of a classical and a quantum code one can associate geometric spaces constructed using methods from noncommutative geometry, arising from rational noncommutative tori and finite abelian group actions on Cuntz algebras and fractals associated to the classical codes. 1.
QUANTUM STABILIZER CODES AND BEYOND
, 2008
"... The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. Despite the large body of literature in quantum coding theory, many important questions, especially those centering on the issue of “good codes” are unresolved. In this dissertat ..."
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Cited by 1 (0 self)
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The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. Despite the large body of literature in quantum coding theory, many important questions, especially those centering on the issue of “good codes” are unresolved. In this dissertation the dominant underlying theme is that of constructing good quantum codes. It approaches this problem from three rather different but not exclusive strategies. Broadly, its contribution to the theory of quantum error correction is threefold. Firstly, it extends the framework of an important class of quantum codes – nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. In particular it provides many explicit constructions of stabilizer codes, most notably it simplifies the criteria by which quantum BCH codes can be constructed from classical codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes.
ON THE THEORY OF Fq-LINEAR Fq t-CODES
, 2013
"... In [7], self-orthogonal additive codes over F4 under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In this paper we consider a natural generalization of additive codes called Fq-linear F q t-codes. We examine a number o ..."
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In [7], self-orthogonal additive codes over F4 under the trace inner product were connected to binary quantum codes; a similar connection was given in the nonbinary case in [33]. In this paper we consider a natural generalization of additive codes called Fq-linear F q t-codes. We examine a number of classical results from the theory of Fq-linear codes, and see how they must be modified to give analogous results for Fq-linear F q t-codes. Included in the topics examined are the MacWilliams Identities, the Gleason polynomials, the Gleason-Pierce Theorem, Mass Formulas, the Balance Principle, the Singleton Bound, and MDS codes. We also classify certain of these codes for small lengths using the theory developed.
