Hybrid oscillator-qudit quantum processors: stabilizer states and symplectic operations

We construct stabilizer states and error-correcting codes on combinations of discrete- and continuous-variable systems, generalizing Gottesman-Kitaev-Preskill (GKP) codes. Our framework [arXiv:2508.04819] absorbs the discrete phase space of a qudit into a hybrid phase space parameterizable entirely by the continuous variables of a harmonic oscillator. The unit cell of a hybrid quantum lattice grows with the qudit dimension, yielding a way to simultaneously measure an arbitrarily large range of non-commuting position and momentum displacements. Simple hybrid states can be obtained by applying a conditional displacement to a Gottesman-Kitaev-Preskill (GKP) state and a Pauli eigenstate, or by encoding some of the physical qudits of a stabilizer state into a GKP code. The states' oscillator-qudit entanglement cannot be generated using symplectic (i.e., Gaussian-Clifford) operations, distinguishing them as a resource from tensor products of oscillator and qudit stabilizer states. As such, these states are relevant to computation with Gaussian-Clifford operations and conditional displacements [arXiv:2412.13164]. We construct general hybrid error-correcting codes by relating stabilizer codes to non-commutative tori and obtaining logical operators via Morita equivalence. We provide examples using commutation matrices, integer symplectic matrices, and binary codes.