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 $\{|1\rangle,\ldots,|d\rangle\}$ and $U$ be an unknown unitary on $\mathcal{H}$. Our protocol estimates $U|d\rangle$ to within trace distance error $\varepsilon$ using $O(\min\{d^{3/2}/\varepsilon,d/\varepsilon^2\})$ forward queries to $U$. This complements the previous result $O(d\log(d)/\varepsilon)$ by van Apeldoorn, Cornelissen, Gily\'en, and Nannicini (SODA 2023), which requires both forward and inverse queries. Moreover, our result implies a query upper bound $O(\min\{d^{3/2}/\varepsilon,1/\varepsilon^2\})$ for inverse-free amplitude estimation, improving the previous best upper bound $O(\min\{d^{2}/\varepsilon,1/\varepsilon^2\})$ based on optimal unitary estimation by Haah, Kothari, O'Donnell, and Tang (FOCS 2023), and disproving a conjecture posed in Tang and Wright (2025).

翻译：本文提出了一种无逆查询的纯量子态估计协议，该协议实现了海森堡标度。具体而言，设 $\mathcal{H}\cong \mathbb{C}^d$ 为一个 $d$ 维希尔伯特空间，其具有标准正交基 $\{|1\rangle,\ldots,|d\rangle\}$，且 $U$ 为 $\mathcal{H}$ 上的未知酉算子。我们的协议以 $O(\min\{d^{3/2}/\varepsilon,d/\varepsilon^2\})$ 次对 $U$ 的正向查询为代价，将 $U|d\rangle$ 估计至迹距离误差 $\varepsilon$ 以内。这补充了 van Apeldoorn、Cornelissen、Gilyén 和 Nannicini（SODA 2023）先前的结果 $O(d\log(d)/\varepsilon)$，该结果要求同时进行正向和逆向查询。此外，我们的结果暗示了无逆查询幅度估计的查询上界为 $O(\min\{d^{3/2}/\varepsilon,1/\varepsilon^2\})$，改进了基于 Haah、Kothari、O'Donnell 和 Tang（FOCS 2023）最优酉估计的先前最佳上界 $O(\min\{d^{2}/\varepsilon,1/\varepsilon^2\})$，并推翻了 Tang 和 Wright（2025）提出的一个猜想。