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.

通过在线学习和阴影层析成像程序，我们研究了适用于量子过程的在线学习任务，尤其着重于具有有界门复杂性和 Pauli 通道的在线学习模型。我们还提供了一种有效的阴影层析成像程序用于 Pauli 通道，并对在线学习的上界和计算下界进行了补充。

量子过程的在线学习

Online learning of quantum processes

Entanglement is a useful resource for learning, but a precise
characterization of its advantage can be challenging. In this work, we consider
learning algorithms without entanglement to be those that only utilize
separable states, measurements, and operations between the main system of
interest and an ancillary system. These algorithms are equivalent to those that
apply quantum circuits on the main system interleaved with mid-circuit
measurements and classical feedforward. We prove a tight lower bound for
learning Pauli channels without entanglement that closes a cubic gap between
the best-known upper and lower bound. In particular, we show that
$\Theta(2^n\varepsilon^{-2})$ rounds of measurements are required to estimate
each eigenvalue of an $n$-qubit Pauli channel to $\varepsilon$ error with high
probability when learning without entanglement. In contrast, a learning
algorithm with entanglement only needs $\Theta(\varepsilon^{-2})$ rounds of
measurements. The tight lower bound strengthens the foundation for an
experimental demonstration of entanglement-enhanced advantages for
characterizing Pauli noise.

展示了不利用纠缠态的学习算法需要 Theta (2^nε^-2) 轮测量来估计 n-qubit 保利（Pauli）通道的每个特征值到 ε 误差，并对不使用纠缠态的学习算法的紧密下界进行了证明。