Sample-efficient estimation of nonlinear quantum state functions

Communications Physics, vol. 9

Abstract

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.

Authors 4

1. Hongshun Yao

   The Hong Kong University of Science and Technology (Guangzhou)

   Affiliation as printed

   Thrust of Artificial Intelligence, Information Hub, Hong Kong University of Science and Technology Guangzhou, Guangdong, China

2. Yingjian Liu Aachen Applied Quantum Algorithms Leiden Instituut-Lorentz

   Leiden University · The Hong Kong University of Science and Technology (Guangzhou)

   Affiliation as printed

   Applied Quantum Algorithms Leiden, Leiden, The Netherlands

   Instituut-Lorentz, Universiteit Leiden, Leiden, RA, The Netherlands

   Thrust of Artificial Intelligence, Information Hub, Hong Kong University of Science and Technology Guangzhou, Guangdong, China

3. Tengxiang Lin

   The Hong Kong University of Science and Technology (Guangzhou)

   Affiliation as printed

   Thrust of Artificial Intelligence, Information Hub, Hong Kong University of Science and Technology Guangzhou, Guangdong, China

4. Xin Wang (91924) corresponding

   The Hong Kong University of Science and Technology (Guangzhou)

   Affiliation as printed

   Thrust of Artificial Intelligence, Information Hub, Hong Kong University of Science and Technology Guangzhou, Guangdong, China