Sample-Efficient Estimation of Nonlinear Quantum State Functions

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 arbitrarily normalized 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 for fundamental tasks, including entropy, fidelity, and eigenvalue estimations. Specifically, for estimating von Neumann entropy, quantum relative entropy, and quantum state fidelity, where $\kappa$ and $\gamma$ represent the minimal nonzero eigenvalue and normalized factor, respectively, we achieve a sample complexity of $\tilde{\mathcal{O}}(\gamma^2/(\varepsilon^2\kappa))$. 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.