How to learn a quantum Hamiltonian?

封闭量子系统的量子动力学由哈密顿量控制。显然，这些哈密顿量通常被假定为已知的，从对手头系统的基本理解中推导出来。然而，尽管这个假设听起来很基本，但在各种看似合理的物理情况下，基本的哈密顿量和伴随的运动方程并不精确。该项目旨在全面探索过去很少受到关注的关键问题：如何从数据中验证，学习和恢复Hamilton？数据驱动恢复的问题最近已经成为人们关注的焦点，这是有充分理由的，尤其是由于量子模拟器的出现，它允许精确监测哈密顿时间演化。它们同时也是量子力学基础的核心兴趣。毕竟，在实验情况下，数据最终就是一切。从这个意义上说，量子力学的表述方式是“错误的”，当假设完全了解一个哈密顿量摆在首位。这个项目是沿着新的想法组织的哈密尔顿证明，哈密尔顿学习和哈密尔顿断层扫描，由知识层次结构的指导下获得。为了实现雄心勃勃的目标，制定了一个跨学科的方法开发计划，建立在超分辨率，凸优化和松弛，张量网络和量子多体动力学的思想基础上。在该项目的最后，一个可靠的机器从数据中学习量子哈密顿。

Quantum dynamics of closed quantum systems is governed by Hamiltonians. These Hamiltonians are obviously usually assumed to be known, derived from a basic understanding of the systems at hand. However, basic as this assumption may sound, in various plausible physical situations, the underlying Hamiltonian and the concomitant equations of motion are not precisely known. This project sets out to comprehensively explore key questions that have received surprisingly little attention in the past: How can a Hamiltonian be certified, learned and recovered from data? Issues of data-driven recovery have for good reasons been moving into the focus of attention very recently, not the least due to the advent of quantum simulators that allow for the precise monitoring of Hamiltonian time evolution. They are of core interest at the same time for the foundations of quantum mechanics. After all, in experimental situations, data is ultimately all there is. In this sense, quantum mechanics is formulated the "wrong way round" when assuming perfect knowledge of a Hamiltonian in the first place. This project is organized along novel ideas of Hamiltonian certification, Hamiltonian learning and Hamiltonian tomography, guided by a hierarchy of knowledge to be acquired. In order for the ambitious aims to be accomplished, an interdisciplinary method development program is laid out, building upon ideas of superresolution, convex optimization and relaxation, tensor networks and quantum many-body dynamics. At the end of the project stands a reliable machinery of robust and scalable learning of quantum Hamiltonians from data.