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 .
URI: http : / /hdl .handle .net /1969 .1 /86011
Date: 2008-10-10

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Quantum stabilizer codes and beyond. Available electronically from http : / /hdl .handle .net /1969 .1 /86011 .

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