The performance of multimode Bosonic quantum error correcting codes

Bosonic error correcting codes encode information into the infinite dimensional Hilbert space of harmonic oscillators. Recently, such codes have seen impressive experimental demonstrations of beyond break-even performance. In a single mode, rotation codes are characterized by a discrete rotation symmetry in their Wigner functions and include codes such as the cat and binomial codes. Meanwhile, the Gottesman-Kitaev-Preskil (GKP) code is characterized by a discrete translation symmetry in its Wigner function. Rotation codes naturally protect against fock shift and dephasing errors, while GKP naturally protects against displacement errors. However, most physical noise processes are dominated by some combination of loss, thermal noise, and dephasing. Understanding the performance of bosonic codes against these relevant channels remains an outstanding challenge. For various regimes of noise strengths, we numerically find which codes perform the best out of existing proposed single and two-mode bosonic codes (e.g. square and hex GKP, binomial and pair-binomial, and cat and pair-cat codes). We furthermore present novel families of multimode rotation codes which offer better performance than existing codes in many of these noise regimes.