My MPO is non-Hermite, it seems that applyExp() function is not suitable, and generalized exponential integrator like "ZGEXPV() function in https://www.maths.uq.edu.au/expokit/fortran/zgexpv.f, or https://github.com/MacroUniverse/Expokit" could work. Does ITensor provide a similar function to solve the problem?
Have you tried epplyExp for non-Hermitian matrix? I don't quite remember the details, but it may be able to handle non-Hermitian matrices. If not, for now you will need to try some of the packages that you mentioned such as Expokit.
In the Julia version, we use KrylovKit.jl (https://github.com/Jutho/KrylovKit.jl). There are also other packages like https://github.com/SciML/ExponentialUtilities.jl which provide that functionality.