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

TL;DR

This paper compares GKP and number-phase bosonic quantum error-correcting codes under realistic noise, identifying the regimes where each code outperforms the other and providing a methodology for optimizing their parameters.

Contribution

It establishes a quantitative performance boundary between GKP and NP codes under photon loss and dephasing noise, guiding code selection based on noise conditions.

Findings

GKP outperforms NP codes when dephasing is much weaker than loss.

The crossover noise regime occurs when dephasing is about two orders of magnitude smaller than loss.

The work offers a practical method for benchmarking and optimizing bosonic codes in realistic environments.

Abstract

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.