Since my question has no answer yet (for about two weeks now) I thought it would be appropriate to include my own conclusions (at the present time ) in the form of an answer.
In Abrams' PhD Thesis (partial, freely available on the Internet ) there is a partial answer to this question, pages 79-80
The Abrams - Lloyd algorithm depends on the possible implementation of a nonlinear OR gate. This nonlinear OR gate can be simulated (imbedded/embedded into, if you like ) by a higher dimensional unitary gate , by setting the extra ancillary qubit(s) to a definite state (that's equivalent to measuring the ancillary qubit(s)). The problem is that the output qubits are entangled, so at the next iteration if I try to set the ancillary input qubits to a definite state (in order to simulate again the nonlinear OR gate) then I collapse the whole system to an eigenstate, I destroy the superposition.
However you could increase the dimensionality of the embedding unitary gate for each iteration (that means increasing the number of input/output qubits , including ancillary qubits ). In other words for the first iteration the nonlinear OR gate will be simulated by a unitary gate $U_1 $ of dimension $n+k$ (where $k$ is a constant), with some ancillary qubits set on input to state $\vert 0 \rangle$ for example. All the $n+k$ output qubits of $U_1$ are entangled. At the next iteration, I simulate $U_1$ using a unitary $U_2$ gate of dimension $n+2k$ , by setting some extra ancillary qubits to $\vert 0 \rangle$. At the next iteration, I simulate $U_2$ using a unitary gate $U_3$ of dimension $n+3k$ , by setting some extra ancillary qubits to $\vert 0 \rangle$. And so on. This way I can simulate the nonlinear OR gate for each iteration and I do not collapse the whole system to an eigenstate at any intermediate stage of the algorithm. We just have to track the right qubits to measure at the end of the algorithm.
In other words, there might be some hope related to the implementation of this algorithm, but I would not make a large bet on it, because simulating $U_1$ $U_2$ , $U_3$ , ..... is not exactly the same thing as simulating the nonlinear OR gate (even if $U_1$ with some ancillary qubits set to $\vert 0 \rangle$ does indeed simulate the nonlinear OR gate). When you set the ancillary qubits of $U_2$ to $\vert 0 \rangle$ you need $U_2$ to simulate $U_1$ when its own ancillary qubits are set to $\vert 0 \rangle $. (in order to simulate the nonlinear OR gate). When you set the ancillary qubits of $U_3$ to $\vert 0 \rangle$ you need $U_3$ to simulate $U_2$ when its own ancillary qubits are set to $\vert 0 \rangle$ , and so on. The problem requires a careful analysis, and at this point I don't know if it allows a solution.
