  We propose a protocol to realize nonadiabatic geometric quantum computation
of small-amplitude Schr\"odinger cat qubits via invariant-based reverse
engineering. We consider a system with a two-photon driven Kerr nonlinearity,
which provides a pair of dressed even and odd coherent states, i.e.,
Schr\"odinger cat states for fault-tolerant quantum computations. An additional
coherent field is applied to linearly drive a cavity mode, to induce
oscillations between dressed cat states. By designing this linear drive with
invariant-based reverse engineering, nonadiabatic geometric quantum computation
with cat qubits can be implemented. The performance of the protocol is
estimated by taking into account the influence of systematic errors, additive
white Gaussian noise, and decoherence including photon loss and dephasing.
Numerical results demonstrate that our protocol is robust against these
negative factors. Therefore, this protocol may provide a feasible method for
nonadiabatic geometric quantum computation in bosonic systems.
