Hybrid stabilizer quantum error correction on abelian configuration spaces

Experimental platforms routinely access Hilbert spaces far beyond qubit subspaces, including harmonic-oscillator (bosonic) and rotor modes that mediate gates or anharmonic modes providing effective logical encodings. While qubit stabilizer codes have received the most attention due to their mature theory, incorporating all the experimentally accessible systems into a unified framework for error correction may lead to lower resource overheads. Toward this end, we define a formalism that generalizes stabilizer quantum error correction to systems with abelian configuration spaces. Our approach unifies qubit/qudit stabilizer codes with bosonic and rotor codes, thus encompassing multi-qubit and multi-qudit stabilizer codes, multi-mode GKP codes, homological rotor codes, and rotation-symmetric bosonic codes (cat, binomial). Within our framework, we introduce a generalized notion of code distance that specializes to Pauli weight in the qubit setting and to Euclidean phase-space distance in the bosonic setting. The formalism makes explicit which experimentally accessible systems can be leveraged for protection, clarifying trade-offs in resource overheads when combining discrete and continuous variables.