Large quantum computers have the potential to vastly outperform any classical computer. The biggest obstacle to building quantum computers of such size is noise. For example, state of the art superconducting quantum computers have average decoherence (loss of information) times of just microseconds. Thus, the field of quantum error correction is especially crucial to progress in the development of quantum technologies. In this research, we study quantum error correction for general noise, which is given by a linear Hermitian map. In standard quantum error correction, the usual assumption is to constrain the errors to completely positive maps, which is a special case of linear Hermitian maps. We establish constraints and sufficient conditions for the possible error correcting codes that can be used for linear Hermitian maps. Afterwards, we expand these sufficient conditions to cover a large class of general errors. These conditions lead to currently known conditions in the limit that the error map becomes completely positive. The later chapters give general results for quantum evolution maps: a set of weak repeated projective measurements that never break entanglement and the asymmetric depolarizing map composed with a not completely positive map that gives a completely positive composition. Finally, we give examples.
| Identifer | oai:union.ndltd.org:siu.edu/oai:opensiuc.lib.siu.edu:dissertations-2898 |
| Date | 01 June 2021 |
| Creators | Gonzales, Alvin Rafer |
| Publisher | OpenSIUC |
| Source Sets | Southern Illinois University Carbondale |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Dissertations |
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