  Shortcuts to adiabaticity are alternative fast processes which reproduce the
same final state as the adiabatic process in a finite or even shorter time,
which have been extended from Hermitian systems to non-Hermitian systems in
recent years, but they are barely explored for general non-Hermitian systems
where off-diagonal elements of the Hamiltonian are not Hermitian. In this
paper, we propose a shortcuts to adiabaticity technique which is based on a
transitionless quantum driving algorithm to realize population transfer for
general two-level non-Hermitian systems and give both exact and approximate
analytical solutions of the corresponding counteradiabatic driving Hamiltonian,
where the latter can be extended to the zeroth-order and first-order terms by
applying perturbative theory. We find that the first-order correction term is
different from the previous results, which is caused by the non-Hermiticity of
the off-diagonal elements. We work out an exact expression for the control
function and present examples consisting of a general two-level system with
gain and loss to show the theory. The results suggest that the high-fidelity
population transfer can be implemented in general non-Hermitian systems by our
method, which works even with strong non-Hermiticity and without rotating wave
approximation. Furthermore, we show that the general Hamiltonian the
off-diagonal elements of which are not conjugate to each other can be
implemented in many physical systems with the present experimental technology,
such as an atom-light interaction system and whispering-gallery microcavity,
which might have potential applications in quantum information processing.
