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Description:
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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 . 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 . |