Adaptive Non-local Observable on Quantum Neural Networks

--Conventional V ariational Quantum Circuits (VQCs) for Quantum Machine Learning typically rely on a fixed Her-mitian observable, often built from Pauli operators. Inspired by the Heisenberg picture, we propose an adaptive non-local measurement framework that substantially increases the model complexity of the quantum circuits. Our introduction of dynamical Hermitian observables with evolving parameters shows that optimizing VQC rotations corresponds to tracing a trajectory in the observable space. This viewpoint reveals that standard VQCs are merely a special case of the Heisenberg representation. Furthermore, we show that properly incorporating variational rotations with non-local observables enhances qubit interaction and information mixture, admitting flexible circuit designs. Two non-local measurement schemes are introduced, and numerical simulations on classification tasks confirm that our approach outperforms conventional VQCs, yielding a more powerful and resource-efficient approach as a Quantum Neural Network. Quantum Machine Learning (QML) is a developing field that leverages the principles of quantum mechanics to advance machine learning (ML) models. With the rapid advancement of quantum computing hardware, QML aims to exploit quantum phenomena--such as superposition, entanglement, and quantum interference--to provide computational advantages over classical approaches. Despite the current limitations of quantum hardware, hybrid quantum-classical algorithms have been developed to harness the strengths of both computing paradigms, allowing near-term quantum devices to contribute meaningfully to real-world ML tasks.