Hybrid Oscillator-Qudit Quantum Processors: stabilizer states, stabilizer codes, symplectic operations, and non-commutative geometry

We construct stabilizer states and error-correcting codes on combinations of discrete- and continuous-variable systems, generalizing the Gottesman-Kitaev-Preskill (GKP) quantum lattice formalism. Our framework 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. Simple hybrid codes can be thought of as subsystem GKP codes whose gauge factor is entangled with a qudit. Our numerical investigations suggest that such codes can sometimes outperform GKP codes against physical noise, and their decoders can be tuned to accommodate either more qudit or more oscillator errors. We also relate stabilizer codes to non-commutative tori, identifying that a general construction of such tori yields multi-mode multi-qudit extensions of GKP codes. We explicitly calculate these codes' logical dimension and logical operators by utilizing the Morita equivalence between their stabilizer and logical tori. We provide examples using commutation matrices, integer symplectic matrices, and binary codes.