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Quantum stabilizer codes and beyond

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Title: Quantum stabilizer codes and beyond
Author: Sarvepalli, Pradeep Kiran
Abstract: 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. These codes are expected to have efficient error recovery schemes than stabilizer codes. Prior to our work however, systematic methods to construct these codes were few and it was not clear how to fairly compare them with other classes of quantum codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work established a close link between subsystem codes and classical codes and it became clear that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels. This approach is based on a Calderbank- Shor-Steane construction that combines BCH and finite geometry LDPC codes.
Publisher: Texas A&M University
Subject: finite geometry codes
Reed-Muller codes
Clifford codes
encoding quantum codes
bounds on quantum codes
subsystem codes
quantum codes
LDPC codes
nonbinary codes
stabilizer codes
asymmetric codes
dual codes
MDS codes
self-orthogonal codes
operator quantum error correction
BCH codes
URI: http://hdl.handle.net/1969.1/86011
Date: 2008-08


Sarvepalli, Pradeep Kiran (2008). Quantum stabilizer codes and beyond. Doctoral dissertation, Texas A&M University. Texas A&M University. Available electronically from http : / /hdl .handle .net /1969 .1 /86011.

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