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!


1 Answer 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.