量子力学可由神经网络实现，揭示了物理与深度学习的深层联系。
Quantum Mechanics and Neural Networks
- 利用数学定理将量子力学映射为神经网络结构
- 数值验证中复现海森堡不确定性与能谱等经典结果
- 适合对量子计算与神经网络交叉领域感兴趣的读者
我们证明任意欧几里得时间量子力学理论均可通过神经网络表示，依据科桑比-卡尔亨-洛夫定理、均方路径连续性及有限两点函数。反射正性（关联幺正性）可通过参数空间分割或马尔可夫性等机制实现。网络非可微性对应非平凡对易子的出现。作用于马尔可夫过程的神经网络虽不再保持马尔可夫性，但仍满足反射正性，从而支持深层神经网络量子系统的构建。我们在多个例子中进行数值实现，复现了海森堡不确定性原理、非平凡对易子及能谱等经典量子力学结果。
原文摘要 · Abstract (English)
We demonstrate that any Euclidean-time quantum mechanical theory may be represented as a neural network, ensured by the Kosambi-Karhunen-Loève theorem, mean-square path continuity, and finite two-point functions. The additional constraint of reflection positivity, which is related to unitarity, may be achieved by a number of mechanisms, such as imposing neural network parameter space splitting or the Markov property. Non-differentiability of the networks is related to the appearance of non-trivial commutators. Neural networks acting on Markov processes are no longer Markov, but still reflection positive, which facilitates the definition of deep neural network quantum systems. We illustrate these principles in several examples using numerical implementations, recovering classic quantum mechanical results such as Heisenberg uncertainty, non-trivial commutators, and the spectrum.
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