Efficient estimation of nonlinear functions of quantum states is crucial for various key tasks in quantum computing, such as entanglement spectroscopy, fidelity estimation, and feature analysis of quantum data. Conventional methods using state tomography and estimating numerous terms of the series expansion are computationally expensive, while alternative approaches based on a purified query oracle impose practical constraints. In this paper, we introduce the quantum state function (QSF) framework by extending the SWAP test via linear combination of unitaries and parameterized quantum circuits. Our framework enables the implementation of arbitrary degree-$n$ polynomial functions of quantum states with precision $\varepsilon$ using $\mathcal{O}(n/\varepsilon^2)$ copies. We further apply QSF for developing quantum algorithms of fundamental tasks, achieving a sample complexity of $\tilde{\mathcal{O}}(1/(\varepsilon^2\kappa))$ for both von Neumann entropy estimation and quantum state fidelity calculations, where $\kappa$ represents the minimal nonzero eigenvalue. Our work establishes a concise and unified paradigm for estimating and realizing nonlinear functions of quantum states, paving the way for the practical processing and analysis of quantum data.

翻译：高效估计量子态的非线性函数对于量子计算中的多项关键任务至关重要，例如纠缠谱分析、保真度估计以及量子数据的特征分析。传统方法采用态层析术并估计级数展开的众多项，计算代价高昂；而基于纯化查询预言机的替代方案则存在实际限制。本文通过结合线性组合酉算子与参数化量子电路扩展SWAP测试，提出了量子态函数框架。该框架能够以精度ε实现任意n次多项式量子态函数，所需副本数为O(n/ε²)。我们进一步应用QSF框架开发了基础任务的量子算法，在冯·诺依曼熵估计和量子态保真度计算中均实现了Õ(1/(ε²κ))的样本复杂度，其中κ表示最小非零本征值。本研究为量子态非线性函数的估计与实现建立了一个简洁统一的范式，为量子数据的实际处理与分析开辟了新途径。