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I'm trying to replicate with qiskit the results of this paper in which basically they implement a quantum version of the Principal Component Analysis applying Quantum Phase Estimation algorithm in order to find eigenvalues and eigenvectors from a covariance matrix.

Everything works fine for the 2x2 covariance matrix case but when they extend the circuit to the 4x4 case it's not very clear how they modify the phase kickback part of the quantum circuit. In the picture that shows the general circuit for QPE, some multi-target controlled-U gates are shown:

In the 2x2 case this can be made using simple *cU3* gates repeated several times as shown in this answer, where the parameters $\theta, \phi, \lambda$ that are needed for the *cU3* gate are calculated using the qiskit function euler_angles_1q directly from the $e^{2\pi i\rho}$ 2x2 matrix where $\rho$ is the normalized covariance matrix:

$\rho=\frac{\sigma}{tr(\sigma)}$

($\sigma$ is the 2x2 covariance matrix for which we want to find the eigenvalues and the eigenvectors)

But for the 4x4 case they provide a circuit that does the phase kickback with several U3 gates

## My question is:

- Since I am a beginner I don't understand how a controlled multi-target gate is decomposed in this chain of gates. Could you please explain me the idea behind this decomposition or give me some resources to start from in order to understand it?
- Is it possible to decompose a multi-target
*cU3*gate using only 2 single target*cU3*gates?

Hi and welcome to Quantum Computing SE. I read the paper you have an issue with. I had some difficulties with it too. See this for some help: https://quantumcomputing.stackexchange.com/questions/9375/cannot-replicate-results-in-article-on-pricing-financial-derivatives-on-ibm-q although it deals with other issue.

– Martin Vesely – 2020-03-13T18:53:59.440A more general approach of constructing $e^{iHt}$ and the controlled version of it can be found in this thread: https://quantumcomputing.stackexchange.com/q/5567/9459

– Davit Khachatryan – 2020-04-04T21:26:48.940