Interacting quantum Hamiltonians are fundamental to quantum computing. Data-based tomography of time-independent quantum Hamiltonians has been achieved, but an open challenge is to ascertain the structures of time-dependent quantum Hamiltonians using time series measurements taken locally from a small subset of the spins. Physically, the dynamical evolution of a spin system under time-dependent driving or perturbation is described by the Heisenberg equation of motion. Motivated by this basic fact, we articulate a physics-enhanced machine-learning framework whose core is Heisenberg neural networks. In particular, we develop a deep learning algorithm according to some physics-motivated loss function based on the Heisenberg equation, which “forces” the neural network to follow the quantum evolution of the spin variables. We demonstrate that, from local measurements, not only can the local Hamiltonian be recovered, but the Hamiltonian reflecting the interacting structure of the whole system can also be faithfully reconstructed. We test our Heisenberg neural machine on spin systems of a variety of structures. In the extreme case in which measurements are taken from only one spin, the achieved tomography fidelity values can reach about $90\%$. The developed machine-learning framework is applicable to any time-dependent systems whose quantum dynamical evolution is governed by the Heisenberg equation of motion.

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具有机器学习的时间相关量子哈密顿的层析成像

相互作用的量子哈密顿量是量子计算的基础。已经实现了时间无关量子哈密顿量的基于数据的断层扫描，但一个开放的挑战是使用从自旋的小子集局部获取的时间序列测量来确定时间相关量子哈密顿量的结构。在物理上，自旋系统在时间相关驱动或扰动下的动力学演化由海森堡运动方程描述。受这一基本事实的启发，我们阐明了一个物理增强的机器学习框架，其核心是海森堡神经网络。特别是，我们根据基于海森堡方程的一些物理驱动的损失函数开发了一种深度学习算法，它“强制”神经网络跟随自旋变量的量子演化。我们证明，从局部测量，不仅可以恢复局部哈密顿量，而且还可以忠实地重建反映整个系统相互作用结构的哈密顿量。我们在各种结构的自旋系统上测试我们的海森堡神经机。在仅从一次旋转中进行测量的极端情况下，实现的断层扫描保真度值可以达到大约$90\%$. 开发的机器学习框架适用于任何量子动力学演化受海森堡运动方程控制的时间相关系统。