Hi, this question might not relate to the coding problems but have more connections to QML.

I have a problem that makes me confused. I did some Unitary operations on quantum states, and I would like to get information on the probability amplitude for each state; however, assuming that all the probability amplitudes are real numbers first, I will get the measurement results which are the squared absolute values of the probability amplitude for each state.

So, can we seek Quantum Measurement as doing a “quadratic activation function” (like f(x) = |x|^2) on the probability amplitude for each state?
Is this feasible to train Neural Networks by seeking Quantum Measurement as an activation function?

Let me try and rephrase your question to see if I understand:

Assuming that you have a quantum circuit that produces quantum states in the computational basis with real and positive coefficients, can a classical neural network learn the associated probability distribution and extrapolate a wavefunction by taking the square root \psi(x) = \sqrt{p(x)}?

I think the use of “activation function” in your original words might be a little off. Generative neural networks are meant to reconstruct exponentially-large probability distributions in compressed formats. Usually this means that, in the architecture for generative models, there needs to be a softmax/sigmoid activation function at the end of the model — that does the work of interpreting the model’s output as a probability distribution.