Learning about physical systems from quantum-enhanced experiments, relying on a quantum memory and quantum processing, can outperform learning from experiments in which only classical memory and processing are available. Whereas quantum advantages have been established for a variety of state learning tasks, quantum process learning allows for comparable advantages only with a careful problem formulation and is less understood. We establish an exponential quantum advantage for learning an unknown $n$-qubit quantum process $\mathcal{N}$. We show that a quantum memory allows to efficiently solve the following tasks: (a) learning the Pauli transfer matrix of an arbitrary $\mathcal{N}$, (b) predicting expectation values of bounded Pauli-sparse observables measured on the output of an arbitrary $\mathcal{N}$ upon input of a Pauli-sparse state, and (c) predicting expectation values of arbitrary bounded observables measured on the output of an unknown $\mathcal{N}$ with sparse Pauli transfer matrix upon input of an arbitrary state. With quantum memory, these tasks can be solved using linearly-in-$n$ many copies of the Choi state of $\mathcal{N}$, and even time-efficiently in the case of (b). In contrast, any learner without quantum memory requires exponentially-in-$n$ many queries, even when querying $\mathcal{N}$ on subsystems of adaptively chosen states and performing adaptively chosen measurements. In proving this separation, we extend existing shadow tomography upper and lower bounds from states to channels via the Choi-Jamiolkowski isomorphism. Moreover, we combine Pauli transfer matrix learning with polynomial interpolation techniques to develop a procedure for learning arbitrary Hamiltonians, which may have non-local all-to-all interactions, from short-time dynamics. Our results highlight the power of quantum-enhanced experiments for learning highly complex quantum dynamics.

翻译：从量子强化实验中学习物理系统, 依靠量子内存和量子处理, 能够从只有古老记忆和处理的实验中学习出优于优于优于优于优于优于优的实验。 虽然为各种州学习任务建立了量子优势, 量子过程的学习只能带来相似的优势, 只有谨慎的问题配制, 并且不那么容易理解。 我们为学习一个未知的 $- qubit 量子进程建立了一个指数优势 $\ mathcal{N} 。 我们显示量量内存可以有效解决以下任务:(a) 学习一个任意的 $\ 量子内传输的 Pauli 矩阵 。 有了量子记忆, 这些任务可以用直径- $n 美元 的直径调调调调值来解析 。