Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks

TL;DR

This paper establishes fundamental limits on quantum state learning with quantum neural networks, showing that avoiding local minima becomes exponentially unlikely as qubit number increases, and characterizes the role of quantum Fisher information.

Contribution

It develops a no-go theorem for quantum state learning with QNNs, revealing inherent challenges and limitations regardless of circuit structure or initialization.

Findings

Probability of avoiding local minima decreases exponentially with qubits.

Local minima curvature relates to quantum Fisher information.

Numerical simulations validate theoretical bounds.

Abstract

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.