Statistical Analysis of Quantum State Learning Process in Quantum Neural
  Networks
Hao-kai Zhang Chenghong Zhu Mingrui Jing Xin Wang
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.