Characterizing a quantum system by learning its state or evolution is a fundamental problem in quantum physics and learning theory with a myriad of applications. Recently, as a new approach to this problem, the task of agnostic state tomography was defined, in which one aims to approximate an arbitrary quantum state by a simpler one in a given class. Generalizing this notion to quantum processes, we initiate the study of agnostic process tomography: given query access to an unknown quantum channel $Φ$ and a known concept class $\mathcal{C}$ of channels, output a quantum channel that approximates $Φ$ as well as any channel in the concept class $\mathcal{C}$, up to some error. In this work, we propose several natural applications for this new task in quantum machine learning, quantum metrology, classical simulation, and error mitigation. In addition, we give efficient agnostic process tomography algorithms for a wide variety of concept classes, including Pauli strings, Pauli channels, quantum junta channels, low-degree channels, and a class of channels produced by $\mathsf{QAC}^0$ circuits. The main technical tool we use is Pauli spectrum analysis of operators and superoperators. We also prove that, using ancilla qubits, any agnostic state tomography algorithm can be extended to one solving agnostic process tomography for a compatible concept class of unitaries, immediately giving us efficient agnostic learning algorithms for Clifford circuits, Clifford circuits with few T gates, and circuits consisting of a tensor product of single-qubit gates. Together, our results provide insight into the conditions and new algorithms necessary to extend the learnability of a concept class from the standard tomographic setting to the agnostic one.

翻译：通过获取量子系统的状态或演化信息来表征其性质，是量子物理学与学习理论中的一个基础性问题，具有广泛的应用前景。近期，作为解决该问题的新途径，研究者定义了不可知状态层析任务，其目标是在给定类别中寻找一个简化量子态以逼近任意未知量子态。本文将这一概念推广至量子过程，首次提出不可知过程层析的研究：在给定对未知量子信道 $Φ$ 的查询访问权限以及已知信道概念类 $\mathcal{C}$ 的条件下，输出一个量子信道，使其在给定误差范围内尽可能逼近 $Φ$，且逼近效果不劣于概念类 $\mathcal{C}$ 中的任何信道。本工作中，我们为这一新任务提出了若干在量子机器学习、量子计量学、经典模拟与误差缓解中的自然应用场景。此外，我们针对多种概念类设计了高效的不可知过程层析算法，包括泡利字符串、泡利信道、量子JUNTA信道、低阶信道以及由 $\mathsf{QAC}^0$ 电路生成的信道类别。我们所采用的核心技术工具是算子与超算子的泡利谱分析。同时，我们证明：通过使用辅助量子比特，任何不可知状态层析算法均可扩展为求解与之兼容的酉算子概念类的不可知过程层析算法，这直接为我们提供了针对Clifford电路、含少量T门的Clifford电路以及由单量子比特门张量积构成的电路的高效不可知学习算法。综合而言，我们的研究结果揭示了将概念类从标准层析场景扩展到不可知场景所需的条件与新算法，为此提供了理论洞见。