Estimating nonlinear functions of quantum states, such as the moment \({{{\rm{tr}}}}({\rho }^{m})\), is of fundamental and practical interest in quantum science and technology. Here we show a quantum-classical hybrid framework to measure them, where the quantum part is constituted by the generalized swap test, and the classical part is realized by postprocessing the result from randomized measurements. This hybrid framework utilizes the partial coherent power of the intermediate-scale quantum processor and, at the same time, dramatically reduces the number of quantum measurements and the cost of classical postprocessing. We demonstrate the advantage of our framework in the tasks of state-moment estimation and quantum error mitigation.

估计量子态的非线性函数，例如矩 \({{{\rm{tr}}}}({\rho }^{m})\)，在量子科学和技术中具有基础和实际意义。在这里，我们展示了一个量子经典混合框架来测量它们，其中量子部分由广义交换测试构成，经典部分通过对随机测量结果进行后处理来实现。这种混合框架利用了中等规模量子处理器的部分相干能力，同时大大减少了量子测量的数量和经典后处理的成本。我们展示了我们的框架在状态矩估计和量子误差缓解任务中的优势。