Quantum computers capable of fault-tolerant operation are expected to provide
provable advantages over classical computational models. However, the question
of whether quantum advantages exist in the noisy intermediate-scale quantum era
remains a fundamental and challenging problem. The root of this challenge lies
in the difficulty of exploring and quantifying the power of noisy quantum
states. In this work, we focus on the complexity of weakly noisy states, which
we define as the size of the shortest quantum circuit required to prepare the
noisy state. To analyze the complexity, we propose a quantum machine learning
(QML) algorithm that exploits the intrinsic-connection property of structured
quantum neural networks. The proposed QML algorithm enables efficiently
predicting the complexity of weakly noisy states from measurement results,
representing a paradigm shift in our ability to characterize the power of noisy
quantum computation