Quantum Error Correction with Reflexive Stabilizer Codes and Cayley Graphs

Long distance communication of digital data, whether through a physical medium or a broadcast signal, is often subjected to noise. To deliver data reliably through noisy communication channels, one must use codes that can detect and correct the particular noise of the channel. For transmission of classical data, error correcting schemes can be as simple as the sending of replicates. For quantum data, and in tandem the development of machines that can process quantum data, quantum error correcting codes must be developed. In addition to a larger set of possible errors, quantum error correcting schemes must contend with other peculiarities of quantum mechanics, such as the no-cloning theorem which can prevent the sending of replicate messages. Stabilizer codes are one family of quantum error correcting codes which can protect and correct errors expressed in terms of the Pauli group, exploiting its group structure and utilizing classical codes and the corresponding duals. We develop and examine a family of quantum stabilizer codes which arise from reflexive stabilizers. Moreover, we provide a mapping from our reflexive stabilizer codes to the well-known CSS codes developed by Calderbank, Shor, and Steane. For the case of a 4-state system we show that these codes can obtain the minimal embedding for code which can correct any flip or phase error. We also provide heuristic algorithms for creating reflexive stabilizer codes starting from the noise of a quantum channel. Furthermore, we show that the problem can be posed in terms of finding maximal Cayley subgraphs with restrictions imposed by the set of potential errors.