  Gottesman-Kitaev-Preskill (GKP) states have been demonstrated to pose
significant advantages when utilized for fault-tolerant all optical
continuous-variable quantum computing as well as for quantum communications
links for entanglement distribution. However interfacing these systems to
long-lived solid-state quantum memories has remained an open problem. Here we
propose an interface between quantum memories and GKP qubit states based on a
cavity-mediated controlled displacement gate. We characterize the quality of
memory-GKP entanglement as a function of cavity parameters suggesting optimal
regimes of operation for high-quality state transfer between either qubit
states. We further extend this protocol to demonstrate the creation of GKP
cluster states by avoiding the requirement of ancillary optical
quadrature-squeezed light. Utilizing post-selected entanglement swapping
operations for GKP qubits, we demonstrate the utility of our protocol for
high-rate entanglement generation between quantum memories. Extensions and
derivatives of our proposal could enable a wide variety of applications by
utilizing the operational trade-offs for qubits encoded in memory and in the
GKP basis.
