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Subsystem codes
- 44th Annual Allerton Conference on Communication, Control, and Computing
, 2006
"... Abstract — We investigate various aspects of operator quantum error-correcting codes or, as we prefer to call them, subsystem codes. We give various methods to derive subsystem codes from classical codes. We give a proof for the existence of subsystem codes using a counting argument similar to the q ..."
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Abstract — We investigate various aspects of operator quantum error-correcting codes or, as we prefer to call them, subsystem codes. We give various methods to derive subsystem codes from classical codes. We give a proof for the existence of subsystem codes using a counting argument similar to the quantum Gilbert-Varshamov bound. We derive linear programming bounds and other upper bounds. We answer the question whether or not there exist [[n, n −2d+2, r> 0, d]]q subsystem codes. Finally, we compare stabilizer and subsystem codes with respect to the required number of syndrome qudits. I.
Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes
, 2006
"... The essential insight of quantum error correction was that quantum information can be protected by suitably encoding this quantum information across multiple independently erred quantum systems. Recently it was realized that, since the most general method for encoding quantum information is to enc ..."
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Cited by 6 (0 self)
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The essential insight of quantum error correction was that quantum information can be protected by suitably encoding this quantum information across multiple independently erred quantum systems. Recently it was realized that, since the most general method for encoding quantum information is to encode it into a subsystem, there exists a novel form of quantum error correction beyond the traditional quantum error correcting subspace codes. These new quantum error correcting subsystem codes differ from subspace codes in that their quantum correcting routines can be considerably simpler than related subspace codes. Here we present a class of quantum error correcting subsystem codes constructed from two classical linear codes. These codes are the subsystem versions of the quantum error correcting subspace codes which are generalizations of Shor’s original quantum error correcting subspace
Subsystem Code Constructions
, 2007
"... A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilize ..."
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Cited by 2 (2 self)
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A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous MDS subsystem codes is established. Another propagation rule is derived that allow one to obtain longer subsystem codes from a given subsystem code.
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.
Encoding Subsystem Codes
, 806
"... In this paper we investigate the encoding of operator quantum error correcting codes i.e. subsystem codes. We show that encoding of subsystem codes can be reduced to encoding of a related stabilizer code making it possible to use all the known results on encoding of stabilizer codes. Along the way w ..."
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In this paper we investigate the encoding of operator quantum error correcting codes i.e. subsystem codes. We show that encoding of subsystem codes can be reduced to encoding of a related stabilizer code making it possible to use all the known results on encoding of stabilizer codes. Along the way we also show how Clifford codes can be encoded. We also show that gauge qubits can be exploited to reduce the encoding complexity. Introduction. In this paper we investigate encoding of subsystem codes. Our main result is that encoding of a subsystem code can be reduced to the encoding of a related stabilizer code, thereby making use of the previous theory on encoding stabilizer codes [2–4]. We shall prove this in two steps. First, we shall show that Clifford codes can be encoded using the same
extraspecial groups
, 2008
"... efficient quantum algorithm for the hidden subgroup problem in ..."
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Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes
, 2006
"... Abstract — The essential insight of quantum error correction was that quantum information can be protected by suitably encoding this quantum information across multiple independently erred quantum systems. Recently it was realized that, since the most general method for encoding quantum information ..."
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Abstract — The essential insight of quantum error correction was that quantum information can be protected by suitably encoding this quantum information across multiple independently erred quantum systems. Recently it was realized that, since the most general method for encoding quantum information is to encode it into a subsystem, there exists a novel form of quantum error correction beyond the traditional quantum error correcting subspace codes. These new quantum error correcting subsystem codes differ from subspace codes in that their quantum correcting routines can be considerably simpler than related subspace codes. Here we present a class of quantum error correcting subsystem codes constructed from two classical linear codes. These codes are the subsystem versions of the quantum error correcting subspace codes which are generalizations of Shor’s original quantum error correcting subspace codes. For every Shor-type code, the codes we present give a considerable savings in the number of stabilizer measurements needed in their error recovery routines. I.
Remarks on Codes, Spectral Transforms, and Decision Diagrams
"... In this paper, we discuss definitions, features, and relationships of Reed-Muller transforms, Reed-Muller codes and their generalizations to multiple-valued cases, and Reed-Muller decision diagrams. The novelty in this primarily review paper resides in putting together these concepts in the same con ..."
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In this paper, we discuss definitions, features, and relationships of Reed-Muller transforms, Reed-Muller codes and their generalizations to multiple-valued cases, and Reed-Muller decision diagrams. The novelty in this primarily review paper resides in putting together these concepts in the same context and providing a uniform point of view to their definition in terms of a convolutionwise multiplication. In particular, we point out that the Plotkin construction schemes for Reed-Muller codes used in coding theory are a different notation for basic Reed-Muller transform matrices over finite fields or can be alternatively viewed as decomposition rules used to define the Reed-Muller decision diagrams. 1
Structures and Constructions of Subsystem Codes over Finite Fields
"... Abstract—Quantum information processing is a rapidly mounting field that promises to accelerate the speed up of computations. The field utilizes the novel fundamental rules of quantum mechanics such as accelerations. Quantum states carrying quantum information are tempted to noise and decoherence, t ..."
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Abstract—Quantum information processing is a rapidly mounting field that promises to accelerate the speed up of computations. The field utilizes the novel fundamental rules of quantum mechanics such as accelerations. Quantum states carrying quantum information are tempted to noise and decoherence, that’s why the field of quantum error control comes. In this paper, we investigate various aspects of the general theory of quantum error control- subsystem codes. Particularly, we first establish two generic methods to derive subsystem codes from classical codes that are defined over finite fields Fq and F q 2. Second, we derive cyclic subsystem codes and using our two methods, we derive all classes of subsystem codes. Consequently, we construct two famous families of cyclic subsystem BCH and RS codes. Cyclic subsystem RS codes are turned out to be Optimal and MDS codes saturating the singleton bound with equality. Third, we demonstrate several methods of subsystem code constructions by extending, shortening and combining given subsystem codes. Finally, we present tables of upper and lower bounds on subsystem codes parameters 1. I.
