  Quantum repeaters are a promising platform for realizing long-distance
quantum communication and thus could form the backbone of a secure quantum
internet, a scalable quantum network, or a distributed quantum computer.
Repeater protocols that encode information in single- or multi-photon states
are limited by transmission losses and the cost of implementing entangling
gates or Bell measurements. In this work, we consider implementing a quantum
repeater protocol using Gottesman-Kitaev-Preskill (GKP) qubits. These qubits
are natural elements for quantum repeater protocols, because they allow for
deterministic Gaussian entangling operations and Bell measurements, which can
be implemented at room temperature. The GKP encoding is also capable of
correcting small displacement errors. At the cost of additional Gaussian noise,
photon loss can be converted into a random displacement error channel by
applying a phase-insensitive amplifier. Here we show that a similar conversion
can be achieved in two-way repeater protocols by using phase-sensitive
amplification applied in the post-processing of the measurement data, resulting
in less overall Gaussian noise per (sufficiently short) repeater segment. We
also investigate concatenating the GKP code with higher level qubit codes while
leveraging analog syndrome data, post-selection, and path-selection techniques
to boost the rate of communication. We compute the secure key rates and find
that GKP repeaters can achieve a comparative performance relative to methods
based on photonic qubits while using orders-of-magnitude fewer qubits.
