The end of Section III in the paper Improved Simulation of Stabilizer Circuits by Aaronson and Gottesman shows how to compute the inner product of stabilizer states $|\psi\rangle$ and $|\varphi\rangle$ represented by sets of generators of their stabilizer group.
Given generators for $|\psi\rangle$ and $|\varphi\rangle$, it suggests to compute the inner product of $U |\psi\rangle$ and $U |\varphi\rangle$, where $U$ is chosen to ensure that $U |\psi\rangle=|0\rangle^{\otimes n}$. It then computes generators for $U |\varphi\rangle$ represented as a tableau and applies Gaussian elimination to this tableau.
I don't understand the following aspects of this algorithm:
- How does the algorithm conclude? (The final steps of the algorithm are not clear to me)
- Why is this algorithm correct? (The reasoning in the paper is not clear to me)
