Mapping of a many-qubit state onto an oscillator using controlled displacements

We extend the controlled displacement interaction between a qubit and a harmonic oscillator to the multi-qubit (qudit) case. We define discrete quadratures of the qudit and show how the qudit state can be displaced in these quadratures controlled by an oscillator quadrature. Using this interaction, a periodic repetition of the state encoded in the qudit, can be deterministically mapped onto the oscillator, which is initialized in a squeezed state. Based on this controlled displacement interaction, we present a full circuit that encodes quantum information in a superposition of qudit quadrature states, and successively prepares the oscillator in the corresponding superposition of approximate Gottesman-Kitaev-Preskill (GKP) states. This preparation scheme is found to be similar to phase estimation, with the addition of a disentanglement gate. Our protocol for GKP state preparation is efficient in the sense, that the set of qubits need only interact with the oscillator through two time-independent interactions, and in the sense that the squeeze factor (in dB) of the produced GKP state grows linearly in the number of qubits used.