Estimating nonlinear functions of quantum states is essential for quantum information processing tasks such as entropy quantification, fidelity estimation, and entanglement spectroscopy. Existing approaches based on quantum state tomography incur exponential resource overhead, while methods relying on purified query access impose significant practical constraints. Here, we introduce the quantum state function framework, which extends the swap test via a linear combination of unitaries and parameterized quantum circuits. The framework estimates any normalized degree-n polynomial function of a quantum state with precision ε using $${{{\mathcal{O}}}}(n/{\varepsilon }^{2})$$ copies, operating directly on identical copies without block-encoding or purified queries. Applied to von Neumann entropy, quantum relative entropy, and state fidelity, it achieves sample complexity $$\widetilde{{{{\mathcal{O}}}}}({\gamma }^{2}/({\varepsilon }^{2}\kappa ))$$, where κ is the minimal nonzero eigenvalue and γ a normalization factor. This framework establishes a unified paradigm for nonlinear quantum state processing, broadening the toolkit for practical quantum data analysis. Estimating nonlinear functions of quantum states is essential for quantum information processing but often computationally expensive. The quantum state function framework estimates a degree-n polynomial of a quantum state with $${{{\mathcal{O}}}}(n/{\varepsilon }^{2})$$ copies without purified query access, enabling practical entropy, fidelity, and eigenvalue estimation.

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A new quantum state function framework is introduced that estimates any normalized degree-n polynomial function of a quantum state with O(n/ε²) copies, without requiring block-encoding or purified query access. By extending the swap test via linear combination of unitaries and parameterized quantum circuits, the approach achieves practical sample complexity for von Neumann entropy, quantum relative entropy, state fidelity, and entanglement spectroscopy — establishing a unified paradigm for nonlinear quantum state processing.

Sample-efficient estimation of nonlinear quantum state functions