Understanding and controlling engineered quantum systems is key to developing practical quantum technology. However, given the current technological limitations, such as fabrication imperfections and environmental noise, this is not always possible. To address these issues, a great deal of theoretical and numerical methods for quantum system identification and control have been developed. These methods range from traditional curve fittings, which are limited by the accuracy of the model that describes the system, to machine learning (ML) methods, which provide efficient control solutions but no control beyond the output of the model, nor insights into the underlying physical process. Here we experimentally demonstrate a ‘graybox’ approach to construct a physical model of a quantum system and use it to design optimal control. We report superior performance over model fitting, while generating unitaries and Hamiltonians, which are quantities not available from the structure of standard supervised ML models. Our approach combines physics principles with high-accuracy ML and is effective with any problem where the required controlled quantities cannot be directly measured in experiments. This method naturally extends to time-dependent and open quantum systems, with applications in quantum noise spectroscopy and cancellation.

理解和控制工程量子系统是开发实用量子技术的关键。然而，考虑到当前的技术限制，例如制造缺陷和环境噪声，这并不总是可能的。为了解决这些问题，人们开发了大量用于量子系统识别和控制的理论和数值方法。这些方法的范围从传统的曲线拟合到机器学习（ML）方法，传统的曲线拟合受到描述系统的模型精度的限制，机器学习（ML）方法提供有效的控制解决方案，但无法控制模型输出之外的内容，也无法深入了解底层物理过程。在这里，我们通过实验演示了一种“灰盒”方法来构建量子系统的物理模型，并用它来设计最优控制。我们报告了优于模型拟合的性能，同时生成酉和哈密顿量，这是标准监督机器学习模型结构中无法获得的数量。我们的方法将物理原理与高精度机器学习相结合，对于无法在实验中直接测量所需受控量的任何问题都有效。这种方法自然地扩展到时间相关和开放的量子系统，并在量子噪声光谱和消除中得到应用。