Stabilizer codes: the continuous, the infinite, and the exotic

Traditional stabilizer codes operate over prime power local-dimensions. For instance, the 5-qubit code and 9-qubit code operate over local-dimension 2. In this presentation we discuss extending the stabilizer formalism using the local-dimension-invariant framework to import stabilizer codes from these standard local-dimensions to other cases. In particular, we show that any of these traditional stabilizer codes can be used for analog continuous-variable codes (continuous and infinite local-dimension), and consider restrictions in phase space (continuous, but finite) and discretized phase space (discrete, but infinite). The combined restriction would correspond to qudit codes. This puts these settings on equivalent footing as traditional stabilizer codes. Following this, using extensions of the 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 ring, so long as the ring has sufficiently large characteristic. This is of theoretical interest and can be of use for systems whose local-dimension is better described by mathematical rings, for which this permits the use of traditional stabilizer codes for protecting their information as well.

This talk is based on the work: https://arxiv.org/abs/2303.17000.