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.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/86011 |
| Date | 10 October 2008 |
| Creators | Sarvepalli, Pradeep Kiran |
| Contributors | Klappenecker, Andreas |
| Publisher | Texas A&M University |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation, text |
| Format | electronic, born digital |
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