# Calculating the Inner Product using Quantum Phase Estimation

I'm following the method laid out in https://arxiv.org/abs/2011.03429 (Page 23 Equations 13-23) to calculate the inner product of two amplitude embedded vectors using Quantum Phase Estimation. I'm able to construct the state, $$\left|\phi\right\rangle = \frac{1}{2}\left[\left|0\right\rangle\left(\left|x\right\rangle+\left|w\right\rangle\right)+\left|1\right\rangle\left(\left|x\right\rangle-\left|w\right\rangle\right)\right]$$ but I had a question,

The paper mentions using Schmidt decomposition to decompose $$\left |\phi \right \rangle$$ into the following state, $$\left |\phi \right \rangle = \frac{-i}{\sqrt{2}}\left(e^{i\gamma}\left |w_+ \right \rangle-e^{-i\gamma}\left |w_- \right \rangle\right)$$ where, $$\left |w_\pm \right \rangle = \frac{1}{2}\left[\left|0\right\rangle\left(\left|x\right\rangle+\left|w\right\rangle\right)\pm i\left|1\right\rangle\left(\left|x\right\rangle-\left|w\right\rangle\right)\right]$$ How can I do that using Qiskit?

I would like to implement this section on NISQ hardware so any suggestions using Qiskit would be appreciated. Please let me know if you need more information and I'll gladly edit my question.

Thank you!

• Hi, welcome to QSE! You might want to split this into two questions Feb 28 at 14:20