Physical realization of nonlinear operators for quantum computers.
Short answer: if you believe quantum mechanics is an accurate description of nature, then since QM is a linear theory, it isn't possible to physically realize nonlinear operations.

As far as we know, the only nonlinear operation in quantum mechanics is that of measurement, but it's of a very restricted kind: measurement only happens at the end of your computation, and it doesn't allow you to amplify exponentially small amplitudes into non-trivial ones (except with negligible probability).

On the other hand, the kinds of nonlinearities discussed by the Abrams and Lloyd paper (as well as others) are generally those that allow you to take a quantum state with exponentially small amplitude, and amplify it to something nontrivial. We don't believe that it is possible to physically realize these nonlinear operations. Why? If it were possible, then you would be able to solve #P problems efficiently! From a complexity point of view, this consequence effectively chucks out the assumption. Being able to solve NP-complete problems efficiently is laughable; to handle #P with ease is a bad joke taken a little too far. The Abrams and Lloyd paper can be understood as an argument against the possibility of nonlinear quantum mechanics.

To try to answer the last part of your question: if there existed nonlinear quantum gates like described in the Abrams and Lloyd paper, then PostBQP could certainly be simulated by their nonlinear quantum computer (because PostBQP = PP by a result of Aaronson, and their nonlinear quantum computer can solve #P-complete problems efficiently).