Clifford gates for Gottesman-Kitaev-Preskill codes in superconducting circuits

The Gottesman-Kitaev-Preskill (GKP) code is one of the most promising bosonic error correcting codes, and has recently been realized both in trapped ion and superconducting circuit experiments. An attractive feature of the GKP code is the fact that all Clifford gates can be implemented with quadratic Hamiltonians, while Pauli measurements can be performed using homodyne detection. We here moreover show that all single-qubit Clifford gates can be performed by simply updating the phase of local oscillators, thus reducing both software and hardware demands. To this end, we present a simple superconducting two-mode coupler that allows all Clifford gates to be executed using a single piece of hardware. We numerically estimate the gate fidelity for both ideal and realistic gate implementations.