Quantum Error-Correcting Codes with a Covariant Encoding

Given some group G of logical gates, for instance the Clifford group, what are the quantum encodings for which these logical gates can be implemented by simple physical operations, described by some physical representation of G? We study this question by constructing a general form of such encoding maps. For instance, we recover that the [[5, 1, 3]] and Steane codes admit transversal implementations of the binary tetrahedral and binary octahedral groups, respectively. For bosonic encodings, we show how to obtain the GKP and cat qudit encodings by considering the appropriate groups, and essentially the simplest physical implementations. We further illustrate this approach by introducing a two-mode bosonic code defined from a constellation of 48 coherent states, for which all single-qubit Clifford gates correspond to passive Gaussian unitaries.