Computing the quantity ⟨ψ|Π|φ⟩

[Fe-r-oz]

Hi, Stefan,

In Trading classical and quantum computational resources section IV:Stabilizer Rank and Classical Simulation of Pauli-based Computation, Page 7 of the paper, Bravyi et. al., notes the previous two approaches, namely of Aaronson et. al and Garcia et. al that address the problem of computing the inner product between stabilizer states ψ, φ. I see that you have used the latter approach for computation of both the magnitude and phase of ⟨ψ|φ⟩. Cool!

Sergey makes a remark that their "technically different and somewhat simpler approach" is more suited for computing the quantity ⟨ψ|Π|φ⟩.

goal: an algorithm for computing a quantity ⟨ψ|Π|φ⟩, where ψ, φ are n-qubit stabilizer states and Π is a projector onto the codespace of some stabilizer code (Bravyi, et. al).

According to Bravyi et. al,. Let $\mathcal{G} ⊂ \mathcal{P^n}$ be an abelian group with t independent generators P₁, P₂, . . . , Pₜ ∈ $\mathcal{G}$. Define a projector Π onto the G-invariant subspace,

$$\Pi=2^{-t}\sum_{P\in\mathcal{G}}P$$

question: Currently, do we have the functionality to compute the quantity ⟨ψ|Π|φ⟩ ? The computation of this quantity comes up quite often in this and the latter papers of Sergey.

I am looking forward to hear your thoughts about this!

Thank you!