  We present a calculation scheme for bit-flip error rates in dissipative cat
qubits due to various perturbations, by developing second-order perturbation
theory on top of a non-linear dissipative Lindbladian. Following this scheme we
derive an analytical expression for the exponentially small bit-flip rate due
to single photon loss, which shows good agreement with numerical simulations.
We also apply our scheme to other perturbations, such as frequency detuning and
Z gate, and find the corresponding bit-flip rates.
