Suppose $|\phi\rangle$ is a stabilizer state represented using stim, and let $x$ and $y$ be computational basis states (i.e., bitstrings). Assume that for some bitstring x, we know that $\langle x|\phi\rangle= e^{iθ}·2^{-r/2}$.
While the magnitude $|⟨y|φ⟩|$ can be efficiently computed using stabilizer techniques, I want to compute the full complex amplitude $⟨y|φ⟩$ — including the phase — for another bitstring $y$.
Although I could compute the full state vector using sim.state_vector() and extract $⟨y|φ⟩$ from it, this approach has exponential time and memory complexity, which I want to avoid.
Is there an efficient way to compute $⟨y|φ⟩$ directly, ideally using stim's tableau representation or related tools? Any insights, algorithms, or references would be greatly appreciated.
