Some Results on Qudit Quantum Error-Correction
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Quantum computing’s seemingly perpetual promise of nearness still has a few hurdles to surmount before it can become a reality. Among these hurdles is that of protection of information from random errors. A potential solution for this challenge is stabilizer codes, which are the analog of classical linear error-correcting codes, however, with an additional axis of error possibility. By and large, quantum computing is discussed in the standard qubit, or two-level, language, however, it is worth considering the case of qudits, or more than two-level systems. Often times results follow simply from some form of algebraic extension: typically group theory or linear algebra. In this work we consider some features that are not immediately apparent from that approach, more often appealing to physical intuition to guide our mathematical ideas, then proving these ideas using the language of qudit operators. Here we consider two particular previously unexplored ideas. The first idea is that of embedding and inscribing of codes into spaces of different sizes than the stabilizer code was originally designed for. Here, we show that all codes can be embedded, and that for infinitely many primes we can in fact guarantee that the distance is at least preserved–a somewhat surprising result. The second idea is, in a way an application of those presented in the first idea, that of turning many stabilizer codes into hybrid codes by taking advantage of relabelling of syndromes and the rapid increase in syndrome space upon using codes in large spaces. Both of these are somewhat useful in their current forms, however, with some additional mathematical work could be turned into potentially very powerful tools for the protection of quantum information. We finish off by, and along the way, mentioning various future directions to carry work on this.
Cite this version of the work
Lane Gunderman (2020). Some Results on Qudit Quantum Error-Correction. UWSpace. http://hdl.handle.net/10012/15600