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My problem is easy to understand, just how to calculate the matrix of phase operator(or phase gate) acts on multi-qubits so that i can perfrom it in quantum circuit on IBM Quantum Experience
just like the controlled-U gate in phase estimation:
for example:
$$U(y)|φ\rangle=\exp(2πiy)|φ\rangle$$
where $|φ\rangle$ is a two- or three-qubit register.
I believe your question is about the phase estimation algorithm. The controlled U-gate in the case of phase estimation algorithm is formed from some unitary operator U. The phase estimation algorithm assumes that you know the unitary U beforehand, and you want to estimate the phase of a particular eigenvalue of U. Also, in this algorithm, you supply an eigenvector of U in |$\psi$>, and you select this eigenvector of U in a way that it corresponds to the eigenvalue of U whose phase you desire to estimate. So, in general, U in your equation results in a pure phase operation only when |$\psi$> is an eigenvector of U. So, in general, U|$\psi$> = $e^{2 \pi i y}$|$\psi$> is ot true for every |$\psi$>.
Regarding how to construct U - that may be motivated by a particular physical situation. For example, quantum algorithms for molecular simulation and energy level calculations use the phase estimation algorithm, so U would be derived from the molecule. See this example in the quantum chemistry library of Q# for simulation of the hydrogen molecule using phase estimation.
The qiskit textbook also gives a detailed example of how to implement the phase estimation algorithm in qiskit. You may find it helpful.
