Estimating the performance boundary of Gottesman-Kitaev-Preskill codes and number-phase codes

Bosonic quantum error-correcting codes encode logical information in a harmonic oscillator, with the Gottesman-Kitaev-Preskill (GKP) and number-phase (NP) codes representing two fundamentally different encoding paradigms. Although both have been extensively studied, it remains unclear under what physical noise conditions (including photon loss and dephasing) one encoding intrinsically outperforms the other. Here we estimate a quantitative performance boundary between GKP and NP codes under general photon loss-dephasing noise. By optimizing code parameters within each encoding family, we identify the noise regimes in which each code exhibits a fundamental advantage. In particular, we find that the crossover occurs when the dephasing strength is approximately two orders of magnitude smaller than the loss strength, revealing a sharp separation between operational regimes. Beyond this specific comparison, our work establishes a practical and extensible methodology for benchmarking bosonic codes and optimizing their parameters, providing concrete guidance for the experimental selection and deployment of bosonic encodings in realistic noise environments.