We study the problem of learning the Hamiltonian of a many-body quantum system from experimental data. We show that the rate of learning depends on the amount of control available during the experiment. We consider three control models: one where time evolution can be augmented with instantaneous quantum operations, one where the Hamiltonian itself can be augmented by adding constant terms, and one where the experimentalist has no control over the system's time evolution. With continuous quantum control, we provide an adaptive algorithm for learning a many-body Hamiltonian at the Heisenberg limit: $T = \mathcal{O}(\epsilon^{-1})$, where $T$ is the total amount of time evolution across all experiments and $\epsilon$ is the target precision. This requires only preparation of product states, time-evolution, and measurement in a product basis. In the absence of quantum control, we prove that learning is standard quantum limited, $T = \Omega(\epsilon^{-2})$, for large classes of many-body Hamiltonians, including any Hamiltonian that thermalizes via the eigenstate thermalization hypothesis. These results establish a quadratic advantage in experimental runtime for learning with quantum control.

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量子控制在多体哈密顿学习中的优势


我们研究了从实验数据中学习多体量子系统的哈密顿量的问题。我们表明，学习速率取决于实验期间可用的控制量。我们考虑三种控制模型：一种可以通过瞬时量子运算来增强时间演化，一种可以通过添加常数项来增强哈密顿量本身，以及实验人员无法控制系统的时间演化。通过连续量子控制，我们提供了一种自适应算法，用于在海森堡极限处学习多体哈密顿量：$T = \mathcal{O}（\epsilon^{-1}）$，其中 $T$ 是所有实验的总时间演变量，$\epsilon$ 是目标精度。这只需要在产品基础上准备产品状态、时间演变和测量。在没有量子控制的情况下，我们证明了学习是标准量子限制的，$T = \Omega（\epsilon^{-2}）$，适用于大类多体哈密顿量，包括任何通过本征态热化假说热化的哈密顿量。这些结果在量子控制学习的实验运行时中建立了二次优势。