Inverse-free quantum state estimation with Heisenberg scaling

In this paper, we present an inverse-free pure quantum state estimation protocol that achieves Heisenberg scaling. Specifically, let mathcal{H}cong mathbb{C}^d be a d-dimensional Hilbert space with an orthonormal basis \{|1rangle,ldots,|drangle\} and U be an unknown unitary on mathcal{H}. Our protocol estimates U|drangle to within trace distance error varepsilon using O\(min\{d^3/2/varepsilon,d/varepsilon²\}\) forward queries to U. This complements the previous result O\(dlog(d\)/varepsilon) by van Apeldoorn, Cornelissen, Gilyén, and Nannicini (SODA 2023), which requires both forward and inverse queries. Moreover, our result implies a query upper bound O\(min\{d^3/2/varepsilon,1/varepsilon²\}\) for inverse-free amplitude estimation, improving the previous best upper bound O\(min\{d²/varepsilon,1/varepsilon²\}\) based on optimal unitary estimation by Haah, Kothari, O'Donnell, and Tang (FOCS 2023), and disproving a conjecture posed in Tang and Wright (2025).