Towards the implementation of a logical CNOT gate between stabilized Gottesman-Kitaev-Preskill (GKP) qubits - Part 1of 2

Bosonic error correction is a promising candidate for the realization of quantum computation in the presence of noise. Recently, the Gottesman-Kitaev-Preskill (GKP) code with logical states realized as grid states encoded in an oscillator was prepared and stabilized using repeated interactions with an ancilla qubit at ETH and Yale. To realize universal computation with the GKP encoding, or to embed the GKP code in a higher-level layer of error correction, a logical two-qubit gate is needed, such that the full Clifford group can be implemented. Here, we focus on the realization of a logical controlled-NOT gate between GKP code words stabilized in a superconducting cavity. In the first part of this two-part series, we discuss the stabilization of a GKP logical qubit and the engineering of an effective quadrature-quadrature interaction in cQED.