Subsystem methods for continuous-variable quantum computing with the Gottesman-Kitaev-Preskill code

We introduce a decomposition of the Hilbert space of continuous-variable quantum mechanics into two separate subsystems, by providing a tensor-product basis for said Hilbert space. One subsystem is 2-dimensional and is interpreted as "logical", while the remaining subsystem is another quantum mode. The decomposition allows one to extract qubits that are compatible with the Gottesman-Kitaev-Preskill code from any continuous-variable state. We will then show that any continuous-variable computational scheme carries qubit information. We argue how, and in which sense, this construction bridges the gap between standard, qubit-based quantum computing and continuous-variable quantum computing. Finally, we apply the decomposition to various situations of interest for continuous-variable quantum computing: in particular, we reveal the logical information carried by continuous-variable cluster states in their idealized, infinite-energy version and in their physical approximations.