Quantum Key Distribution over quantum repeater chains based on the hexagonal Gottesman-Kitaev-Preskill Code

Quantum communication offers the possibility of generating secure cryptographic key over long distances, but the outstanding challenge is photon loss in optical fibers. To address this concern, repeaters utilizing both heralded and error-correcting schemes are studied. In error-correction, a particular class of bosonic codes provides strong resistance to photon loss errors. These are the Gottesman-Kitaev-Preskill (GKP) codes, of which the square and hexagonal lattice codes have been thoroughly examined. Previous works demonstrated how the square GKP code can be used for communication, leveraging analog syndrome data for post-selection and for concatenation with discrete-variable codes. Alternatively, the hexagonal code was shown to produce greater fidelity, and to heuristically correspond to an optimal encoding. However, it is not known how hexagonal GKP performs in repeater-based architectures in generating secret key rate relative to the square lattice based code. To address this gap, we consider the task of Quantum Key Distribution performed over a repeater chain based on the hexagonal lattice GKP code, comparing the secret key rate and achievable distance with the secret key generation of the square based code. We show that hexagonal GKP generally outperforms the square code both in secret key rate and achievable distance, examine how the hexagonal code performs in imperfect hardware, and analyze the benefit of techniques such as post-selection and advantage distillation in this context.