Quantum machine learning (QML) seeks to exploit the intrinsic properties of quantum mechanical systems, including superposition, coherence, and quantum entanglement for classical data processing. However, due to the exponential growth of the Hilbert space, QML faces practical limits in classical simulations with the state-vector representation of quantum system. On the other hand, phase-space methods offer an alternative by encoding quantum states as quasi-probability functions. Building on prior work in qubit phase-space and the Stratonovich-Weyl (SW) correspondence, we construct a closed, composable dynamical formalism for one- and many-qubit systems in phase-space. This formalism replaces the operator algebra of the Pauli group with function dynamics on symplectic manifolds, and recasts the curse of dimensionality in terms of harmonic support on a domain that scales linearly with the number of qubits. It opens a new route for QML based on variational modelling over phase-space.

翻译：量子机器学习（QML）旨在利用量子力学系统的内在特性——包括叠加、相干性和量子纠缠——来处理经典数据。然而，由于希尔伯特空间的指数级增长，采用量子系统的态矢量表示进行经典模拟时，QML面临着实际限制。另一方面，相空间方法通过将量子态编码为准概率函数，提供了一种替代方案。基于先前在量子位相空间及Stratonovich-Weyl（SW）对应关系上的工作，我们为单量子位与多量子位系统在相空间中构建了一个封闭、可组合的动力学形式体系。该形式体系将泡利群的算子代数替换为辛流形上的函数动力学，并将维度灾难重新表述为定义域上的谐波支撑问题，该定义域的规模随量子位数呈线性增长。这为基于相空间变分建模的QML开辟了一条新路径。