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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:

enter image description here

for example:

$$U(y)|φ\rangle=\exp(2πiy)|φ\rangle$$

where $|φ\rangle$ is a two- or three-qubit register.

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  • $\begingroup$ Welcome to QCSE! The way it's currently defined, $U$ is identity up to global phase, so controlled-$U$ is just a single-qubit $Z$ rotation on the control qubit. Is this really the operation you have in mind? $\endgroup$ – Adam Zalcman Feb 14 at 17:44
  • $\begingroup$ @Adam Zalcman thanks a lot for viewing this The operation U in my mind is the one defined in Phase Estimation,I know that normal U gate can change the phase of a single-qubit, and I want to find a way to define 'U' gate that can change the phase of a quantum state contains multi-qubit,such as |000>,|010> and so on. Simply apply U gate on each single-qubit doesn't work cause the phase on single-qubit will also join the tensor product process. I think it's possible to find the matrix of such operator use linear algebra knowledge but so far I haven't find a way $\endgroup$ – leafkoi Feb 15 at 2:24
  • $\begingroup$ I'm still not clear what it is you're asking for. Are you wanting to know how to implement controlled-$e^{i\theta}U$ instead of controlled-$U$? $\endgroup$ – DaftWullie Mar 12 at 8:48
  • $\begingroup$ You can check the answer given by Dhruv B, it's may be easier to understand. Several days ago I'm trying to verify a quantum program's correctness in Qiskit, and one step of it needs a gate to change the phase of quantum register which contains two qubits at least, that's the birth of this question . I review some books about linear algebra knowledge and found the solution, just treat exp(2πiy) as eigenvalue and |φ⟩ as its corresponding eigenvector,then the matrix is easy to get.@DaftWullie $\endgroup$ – leafkoi Mar 13 at 9:33
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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.

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