Quantum neural networks (QNNs) have been a promising framework in pursuing near-term quantum advantage in various fields, where many applications can be viewed as learning a quantum state that encodes useful data. As a quantum analog of probability distribution learning, quantum state learning is theoretically and practically essential in quantum machine learning. In this paper, we develop a no-go theorem for learning an unknown quantum state with QNNs even starting from a high-fidelity initial state. We prove that when the loss value is lower than a critical threshold, the probability of avoiding local minima vanishes exponentially with the qubit count, while only grows polynomially with the circuit depth. The curvature of local minima is concentrated to the quantum Fisher information times a loss-dependent constant, which characterizes the sensibility of the output state with respect to parameters in QNNs. These results hold for any circuit structures, initialization strategies, and work for both fixed ansatzes and adaptive methods. Extensive numerical simulations are performed to validate our theoretical results. Our findings place generic limits on good initial guesses and adaptive methods for improving the learnability and scalability of QNNs, and deepen the understanding of prior information's role in QNNs.

翻译：量子神经网络（QNNs）已成为在各领域追求近期量子优势的一个重要框架，其中许多应用可视为学习编码有用数据的量子态。作为概率分布学习的量子对应，量子态学习在量子机器学习中具有理论和实践上的重要性。本文提出了一个"不可能定理"，论证了即使从高保真初始态出发，使用QNNs学习未知量子态仍存在根本性局限。我们证明：当损失值低于临界阈值时，避开局部极小值的概率随量子比特数呈指数衰减，仅随电路深度呈多项式增长。局部极小值的曲率集中为量子Fisher信息乘以损失相关常数，该常数表征QNNs中输出态对参数的敏感度。这些结果适用于任意电路结构、初始化策略，并同时适用于固定拟设与自适应方法。我们通过大量数值模拟验证了理论结果。本研究成果为通过良好初始猜测和自适应方法提升QNNs可学习性与可扩展性设立了一般性限制，并深化了对先验信息在QNNs中作用的理解。