用相空间方法突破量子机器学习的维度瓶颈
Quantum Machine Learning in Multi-Qubit Phase-Space Part I: Foundations
- 将量子态表示为准概率函数，替代传统算符代数
- 相空间维度线性增长，避免希尔伯特空间指数膨胀
- 适合研究多比特量子系统，为变分量子建模提供新框架
量子机器学习（QML）旨在利用量子系统的叠加、相干和纠缠等特性处理经典数据。然而，由于希尔伯特空间随比特数指数增长，基于态矢量表示的经典模拟面临实际限制。相空间方法通过将量子态编码为准概率函数提供了替代路径。基于前期在比特相空间与斯特拉顿维奇-韦尔对应关系的研究，本文构建了单比特与多比特系统的封闭、可组合的动力学形式化体系。该形式化将泡利群算符代数替换为辛流形上的函数动力学，并将维数灾难转化为在随比特数线性扩展的域上谐波支持的问题。这一方法为基于相空间的变分建模开辟了新途径。
原文摘要 · Abstract (English)
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.
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