Among recent insights into learning quantum states, online learning and shadow tomography procedures are notable for their ability to accurately predict expectation values even of adaptively chosen observables. In contrast to the state case, quantum process learning tasks with a similarly adaptive nature have received little attention. In this work, we investigate online learning tasks for quantum processes. Whereas online learning is infeasible for general quantum channels, we show that channels of bounded gate complexity as well as Pauli channels can be online learned in the regret and mistake-bounded models of online learning. In fact, we can online learn probabilistic mixtures of any exponentially large set of known channels. We also provide a provably sample-efficient shadow tomography procedure for Pauli channels. Our results extend beyond quantum channels to non-Markovian multi-time processes, with favorable regret and mistake bounds, as well as a shadow tomography procedure. We complement our online learning upper bounds with mistake as well as computational lower bounds. On the technical side, we make use of the multiplicative weights update algorithm, classical adaptive data analysis, and Bell sampling, as well as tools from the theory of quantum combs for multi-time quantum processes. Our work initiates a study of online learning for classes of quantum channels and, more generally, non-Markovian quantum processes. Given the importance of online learning for state shadow tomography, this may serve as a step towards quantum channel variants of adaptive shadow tomography.

翻译：在近期关于量子态学习的见解中，在线学习与影子层析方法因其能够准确预测自适应选定观测量之期望值而备受关注。与量子态情形不同，具有类似自适应特性的量子过程学习任务却鲜少被探讨。本研究致力于探究量子过程的在线学习任务。尽管一般量子通道的在线学习并不可行，我们证明了具有有限门复杂度的通道以及泡利通道可以在在线学习的遗憾界与错误界模型下进行在线学习。事实上，我们能够在线学习任意指数大规模已知通道的概率混合。我们还为泡利通道提供了一个可证明样本高效的影子层析方法。我们的结果不仅适用于量子通道，更可扩展至非马尔可夫多时间过程，并具有优越的遗憾界与错误界，以及相应的影子层析方法。我们通过错误下界与计算下界对在线学习上界进行了补充。在技术层面，我们利用了乘性权重更新算法、经典自适应数据分析、贝尔采样，以及多时间量子过程的量子梳理论工具。本工作开启了对量子通道类别乃至更广义非马尔可夫量子过程在线学习的研究。鉴于在线学习在量子态影子层析中的重要性，这或许能成为迈向自适应影子层析的量子通道变体的关键一步。