| dc.description.abstract | Quantum error-correcting codes play a vital role in protecting quantum information from external noise and internal decoherence, and will be a critical tool in the construction of a universal fault-tolerant quantum computer. Most known quantum error-correcting codes belong to the well-studied class of quantum stabilizer codes, which have a well-behaved group structure that makes them easy to construct and practical to use. In this dissertation, we expand the theory of two generalizations of stabilizer codes: quantum-classical hybrid codes and nonadditive quantum codes.
Hybrid codes simultaneously encode both quantum and classical information together, which allows for some nontrivial advantage over coding schemes with separate transmission. As many quantum communications protocols involve both quantum and classical information, hybrid codes may be useful in designing more efficient schemes. We construct the first known families of genuine hybrid codes that are guaranteed to provide an advantage over quantum stabilizer codes, giving infinite families of both single-error detecting and correcting hybrid codes. We also generalize hybrid codes to allow for differing levels of protection of errors, and give a general construction of hybrid codes of this type from quantum subsystem codes. When used in conjunction with the class of Bacon-Casaccino subsystem codes, this provides for a method of constructing hybrid codes from pairs of classical linear codes. As an application of hybrid codes, we show how they can be used to protect against faulty syndrome measurement errors and inspire the construction of new quantum data-syndrome codes.
Finally, we investigate the Shor-Laflamme weight enumerators for both hybrid and nonadditive quantum codes. Weight enumerators are powerful tools that allow for the construction of linear-programming bounds on the parameters of quantum codes and let us rule out the existence of certain codes. In particular, we show that the weight enumerators of the nonadditive codeword stabilized quantum codes have a combinatorial interpretation analogous to that of quantum stabilizer codes, showing that they may be viewed as the distance enumerators of associated classical codes. | |
