Sarvepalli, Pradeep Kiran (2008-08). Quantum stabilizer codes and beyond. Doctoral Dissertation. Thesis uri icon

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.
  • 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.

publication date

  • August 2008