The performance of random Bosonic rotation codes

Bosonic error correcting codes utilize the infinite dimensional Hilbert space of a harmonic oscillator to encode a qubit and have seen experimental demonstrations of break even performance. Bosonic rotation codes are characterized by a discrete rotation symmetry in their Wigner functions and include codes such as the cat and binomial codes. Rotation codes can naturally protect against small loss and dephasing errors, which are the dominant sources of noise in physical quantum harmonic oscillators. We define several different notions of random bosonic rotation codes and numerically explore their performance against loss and dephasing. We find that random rotation codes can, on average, outperform cat codes. Furthermore, we find the best random rotation codes can outperform binomial codes for simultaneous loss and dephasing errors.