Scalable quantum error correction with the bosonic GKP code

We review the bosonic GKP (Gottesman-Kitaev-Preskill) code which encodes a qubit into an oscillator and its possible implementation in a microwave mode in circuit-QED hardware. We discuss how GKP code states can be created from Schroedinger cat states or from a dispersive interaction with a qubit. We propose a scalable architecture which uses a surface code on top of the GKP qubits. For a noise model of Gaussian stochastic displacement errors, we discuss how to decode such toric-GKP code and give estimates for the threshold standard deviation, corresponding to a low (4 or more) number of average photons in the GKP code states.