Matching lower asymptotic for tangent-instrument cost

Prove a matching lower asymptotic for the tangent instrument’s preparation cost, complementing the established upper bound of order \(\sqrt{\log(1/\epsilon)}\) for preparing the uniform ensemble of real-amplitude pure states under bounded Hamiltonian feedback.

Background

For the tangent monitoring instrument $(\delta=0)$, the paper constructs a first-hit feedback strategy whose required peak Hamiltonian strength grows no faster than $O[\sqrt{\log(1/\epsilon)}]$ as the terminal ensemble error $\epsilon$ tends to zero. Unlike the positive-normal-noise instruments, for which an inverse-accuracy lower bound is proved, the paper does not establish that the tangent upper bound is asymptotically optimal. A matching lower bound would characterize the exact accuracy scaling of the tangent case.