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 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. 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.

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Quantum computers will need effective error-correcting codes. Current quantum processors require precise control of each particle, so having fewer particles to control might be beneficial. Although traditionally quantum computers are considered as using qubits (2-level systems), qudits (systems with more than 2-levels) are appealing since they can have an equivalent computational space using fewer particles, meaning fewer particles need to be controlled. In this work we prove how to construct codes with parameters [[2 N , 2 N − 1 − 2N, ≥ 3]]q for any choice of prime q and natural number N . This is accomplished using the technique of local-dimension-invariant (LDI) codes. Generally LDI codes have the drawback of needing large local-dimensions to ensure the distance is at least preserved, and so this work also reduces this requirement by utilizing the structure of CSS codes, allowing for the aforementioned code family to be imported for any local-dimension choice.

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