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I'm brand new to quantum computing and have been learning Qiskit for a few weeks now. I am attempting to find the number of 0 eigenvalues of the following operator: $$ T = PHP$$ where $P$ is a projection operator and $H$ is hermitian. For example, the projector operator can look like this: $$\begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ \end{pmatrix}.$$ Particularly, $P$ can be constructed by preparing the state $|\psi\rangle = |01\rangle + |10\rangle$ (omitting the constant in front). Using CNOTs to copy $|\psi\rangle$ into another register of qubits to make $|\psi\rangle|\psi\rangle$, I can trace out the first register and have the operator $P = \left| 01\rangle\langle 01\right| + \left| 10\rangle\langle 10\right|$. (Please correct me if this is incorrect).

I also have $H$ written down as a Hermitian matrix. What I'm struggling with is a Qiskit implementation that addresses the following:

  1. How to apply the projector to $H$ from both sides to obtain $T$?
  2. How to retrieve the eigenvalues of $T$ (specifically, I need the algebraic multiplicity of 0-eigenvalues)?

If it helps, $H$ is always real and a $2\times2$ version looks like $$\begin{pmatrix} 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1\\ 1 & 0 & 0 & -1\\ 0 & 1 & -1 & 0\\ \end{pmatrix}.$$ To be clear, I'm looking to do all the above inside of the quantum circuit. So I can't make the operator $P$ and the operator $H$ using something like numpy, multiply them together as matrices, and then put the result into a quantum circuit to find the eigenvalues. Everything needs to happen within in the circuit.

Thanks in advance for any advice or help!

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I'm not super experienced with quantum either, but I'm not sure that what your looking for is doable with in the circuit using a matrix representation. I think if you vectorize the matrices you should be able to apply CNOTs recursively to re-register the qbits.

From there you should be able to use EigsQPE to solve for the eigenvalue multiplicity. (Finding Eigen Values from Quantum Phase Estimation - Using qiskit)

Best of Luck

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