Gottesman-Kitaev-Preskill State Preparation Using Periodic Driving

The Gottesman-Kitaev-Preskill (GKP) code may be used to overcome noise in continuous variable quantum systems. However, preparing GKP states remains experimentally challenging. We propose a method for preparing GKP states by engineering a time-periodic Hamiltonian whose Floquet states are GKP states. This Hamiltonian may be realized in a superconducting circuit comprising a SQUID shunted by a superinductor and a capacitor, with a characteristic impedance twice the resistance quantum. The GKP Floquet states can be prepared by adiabatically tuning the frequency of the external magnetic flux drive. We predict that highly squeezed $>11.9\mathrm{dB}$ (10.8 dB) GKP magic states can be prepared on a microsecond timescale, given a quality factor of ${10}^6$ (${10}^5$) and flux noise at typical rates.