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:)
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:
https://qiskit.org/documentation/stubs/qiskit.algorithms.HamiltonianPhaseEstimation.html
and here:
https://quantum-computing.ibm.com/composer/docs/iqx/guide/quantum-phase-estimation
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
and:
https://qiskit.org/documentation/stubs/qiskit.opflow.primitive_ops.PauliSumOp.exp_i.html
Quantum Principal Component Analysis uses what the authors call a quantum phase algorithm. This refers to the HHL algorithm, not quantum phase estimation which requires a copy of an eigenvector to find its eigenvalue. HHL, which actually use QPE as a subroutine, instead gives the eigenvectors and eigenvalues of a unitary without additional information. In the case of QPCA, this unitary is the exponentiated density matrix containing the data we wish to extract the principal components from.
This video does a good job explaining the HHL algorithm. The wikipedia page also gives a good overview.
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!
