Tight bound for estimating expectation values from a system of linear equations

The System of Linear Equations Problem (SLEP) is specified by a complex invertible matrix $A$, the condition number $\kappa$ vector $b$, Hermitian $M$ and an accuracy $\epsilon$, task to estimate $x^\dagger Mx$, where $x$ solution equation $Ax = b$. We aim establish lower bound on complexity end-to-end quantum algorithms for SLEP with respect devise algorithm that saturates this bound. To make bounds attainable, we consider query in setting which block encoding given, i.e., unitary black box $U_M$ contains $M/\alpha$ as some $\alpha \in \mathbb R^+$. show $\Theta(\alpha/\epsilon)$. Our established reducing problem estimating mean function SLEP. $\Theta(\alpha/\epsilon)$ result tightens proves common assertion polynomial dependence (poly$(1/\epsilon)$) SLEP, shows improvement beyond linear not possible if provided via encoding.