I have been learning Quantum PCA recently but I came across a problem. How can I get the eigenvalues of a matrix by phase estimation? And what on what kind of matrix can we implement this algorithm?(Unitary matrix only?)

Sorry for my poor description:)

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    $\begingroup$ Matrix with eigenvalues $\lambda$ of the form $e^{i\theta}$, i.e., modulus one. Or if you know $|\lambda|$ already. $\endgroup$
    – narip
    May 20, 2022 at 2:06
  • $\begingroup$ Have a look at HHL algorithm. The evaluation of eigenvalues is part of it. $\endgroup$ May 20, 2022 at 6:17

2 Answers 2


The Quantum Phase Estimation algorithm allows you to estimate the phases $\theta_k$ in the eigenvalue equation:

$$ U|\psi_k\rangle = e^{2\pi i \theta_k} |\psi_k\rangle $$

where $|\psi_k\rangle$ and $e^{2\pi i \theta_k}$ correspond to the eigenvectors and eigenvalues of $U$. As you need to implement $U$ in a quantum circuit, this matrix must be unitary. However, you can get the eigenvalues of a Hermitian matrix H as well (which is not necessarily unitary) by doing the exponential of the matrix, i.e:

$$ U = e^{ibH}$$

$b$ is a scale factor that ensures that all the phases $\theta_k$ fall in the interval $[0,1]$ (since what you will measure in the ancillary register is the state $|2^n \theta_k \rangle $ where n is the number of qubits ancilla). You can find more information here:


and here:


If interested, there is also the function 'exp_i()' in Qsikit that allows you to obtain the exponential form of a weighted sum of Pauli operators:

Section 'Evolutions, exp_i(), and the EvolvedOp' of https://qiskit.org/documentation/tutorials/operators/01_operator_flow.html




You may use Quantum Phase estimation to calculate the eigenvalues.here For this algorithm, you need to know the eigenvectors (or have an educated guess).

I hope it help you!

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