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Physics Letters A

Two-stage behavior of mutual information in quantum neural networks

Xin Zhang, Xiaoyu Li, Zhaoyang Dong, Yuexian Hou

Published Oct 30, 2025• 2 citations
Publisher
DOI
10.1016/j.physleta.2025.131110

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Quantum neural networks (QNNs) is a parameterized quantum circuit model, which can be trained by gradient-based optimizer, can be used for supervised learning, regression tasks, combinatorial optimization, etc. Although many works have demonstrated that QNNs have better learnability, generalizability, etc. compared to classical neural networks. However, as with classical neural networks, we still can't explain their working mechanism well. In this paper, we reveal the training mechanism of QNNs by mutual information. Unlike traditional mutual information in neural networks, due to quantum computing remains information conserved, the mutual information is trivial of the input and output of U operator. In our work, in order to observe the change of mutual information during training, we divide the quantum circuit (U operator) into two subsystems, discard subsystem (D) and measurement subsystem (M) respectively. We calculate two mutual information, I(Di : Mo) and I(Mi : Mo) (i and o means input or output of the corresponding subsystem), and observe their behavior during training. As the epochs increases, I(Di : Mo) gradually increases, this may means some information of discard subsystem is continuously pushed into the measurement subsystem, the information should be label-related. What's more, I(Mi : Mo) exist two-phase behavior in training process, this consistent with the information bottleneck anticipation. The first phase, I(Mi : Mo) is increasing, this means the measurement subsystem perform feature fitting. The second phase, I(Mi : Mo) is decreasing, this may means the system is generalizing, the measurement subsystem discard label-irrelevant information into the discard subsystem as many as possible. Our work discussed the working mechanism of QNNs by mutual information, further, it can be used to analyze the accuracy and generalization of QNNs.
The quantum neural network is one of the promising applications for near-term noisy intermediate-scale quantum computers. A quantum neural network distills the information from the input wave function into the output qubits. In this Letter, we show that this process can also be viewed from the opposite direction: the quantum information in the output qubits is scrambled into the input. This observation motivates us to use the tripartite information-a quantity recently developed to characterize information scrambling-to diagnose the training dynamics of quantum neural networks. We empirically find strong correlation between the dynamical behavior of the tripartite information and the loss function in the training process, from which we identify that the training process has two stages for randomly initialized networks. In the early stage, the network performance improves rapidly and the tripartite information increases linearly with a universal slope, meaning that the neural network becomes less scrambled than the random unitary. In the latter stage, the network performance improves slowly while the tripartite information decreases. We present evidences that the network constructs local correlations in the early stage and learns large-scale structures in the latter stage. We believe this two-stage training dynamics is universal and is applicable to a wide range of problems. Our work builds bridges between two research subjects of quantum neural networks and information scrambling, which opens up a new perspective to understand quantum neural networks.
Abstract We propose a method of quantum machine learning called quantum mutual information neural estimation (QMINE) for estimating von Neumann entropy and quantum mutual information, which are fundamental properties in quantum information theory. The QMINE proposed here basically utilizes a technique of quantum neural networks (QNNs), to minimize a loss function that determines the von Neumann entropy, and thus quantum mutual information, which is believed more powerful to process quantum datasets than conventional neural networks due to quantum superposition and entanglement. To create a precise loss function, we propose a quantum Donsker-Varadhan representation (QDVR), which is a quantum analog of the classical Donsker-Varadhan representation. By exploiting a parameter shift rule on parameterized quantum circuits, we can efficiently implement and optimize the QNN and estimate the quantum entropies using the QMINE technique. Furthermore, numerical observations support our predictions of QDVR and demonstrate the good performance of QMINE.