I am trying to understand whether a linear system that is originally defined in terms of statistical estimators can be reformulated into a fully quantum algorithm that avoids measurement-based estimation altogether. More concretely, suppose I have a linear system $A x = b,$ where the matrix $A$ is not given explicitly but is defined through empirical estimators, e.g. expectation values of observables or sample averages: $ A_{ij} = \mathbb{E}[H_{ij}], $ or more generally via some statistical proc