Stabilizer Codes with Exotic Local-dimensions

Lane G. Gunderman

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Abstract

Traditional stabilizer codes operate over prime power local-dimensions. In this work we extend the stabilizer formalism using the local-dimension-invariant setting to import stabilizer codes from these standard local-dimensions to other cases. In particular, we show that any traditional stabilizer code can be used for analog continuous-variable codes, and consider restrictions in phase space and discretized phase space. This puts this framework on an equivalent footing as traditional stabilizer codes. Following this, using extensions of prior ideas, we show that a stabilizer code originally designed with a finite field local-dimension can be transformed into a code with the same $n$, $k$, and $d$ parameters for any integral domain. This is of theoretical interest and can be of use for systems whose local-dimension is better described by mathematical rings, which permits the use of traditional stabilizer codes for protecting their information as well.

This work provides an extension of the traditional stabilizer formalism for encoding quantum information to other settings such as a continuous-variable systems (conjugate quadratures), discretized phase-space, phase encodings, as well as more mathematical settings that may correspond to physical settings.

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