  Nonadiabatic geometric quantum computation (NGQC) has emerged as an excellent
proposal for achieving fast and robust quantum control against control errors.
However, previous NGQC protocols could not be strongly resilient against the
noise from decay of bare states in a realistic system, which can be
equivalently described by a non-Hermitian Hamiltonian. Here, we show how to
perform NGQC in non-Hermitian quantum systems. By utilizing a novel geometric
phase generated by non-unitary evolution of the system, a universal set of
geometric gates can be realized with a high fidelity. Moreover, we demonstrate
that the nonadiabatic process does not lead to the loss of fidelity from decay.
