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.

We consider the problem of learning low-degree quantum objects up to $\varepsilon$-error in $\ell_2$-distance. We show the following results: $(i)$ unknown $n$-qubit degree-$d$ (in the Pauli basis) quantum channels and unitaries can be learned using $O(1/\varepsilon^d)$ queries (independent of $n$), $(ii)$ polynomials $p:\{-1,1\}^n\rightarrow [-1,1]$ arising from $d$-query quantum algorithms can be classically learned from $O((1/\varepsilon)^d\cdot \log n)$ many random examples $(x,p(x))$ (which implies learnability even for $d=O(\log n)$), and $(iii)$ degree-$d$ polynomials $p:\{-1,1\}^n\to [-1,1]$ can be learned through $O(1/\varepsilon^d)$ queries to a quantum unitary $U_p$ that block-encodes $p$. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded~polynomials.

Here we revisit one of the prototypical tasks for characterizing the structure of noise in quantum devices: estimating every eigenvalue of an $n$-qubit Pauli noise channel to error $\epsilon$. Prior work (Chen et al., 2022) proved no-go theorems for this task in the practical regime where one has a limited amount of quantum memory, e.g. any protocol with $\le 0.99n$ ancilla qubits of quantum memory must make exponentially many measurements, provided it is non-concatenating. Such protocols can only interact with the channel by repeatedly preparing a state, passing it through the channel, and measuring immediately afterward. This left open a natural question: does the lower bound hold even for general protocols, i.e. ones which chain together many queries to the channel, interleaved with arbitrary data-processing channels, before measuring? Surprisingly, in this work we show the opposite: there is a protocol that can estimate the eigenvalues of a Pauli channel to error $\epsilon$ using only $O(\log n/\epsilon^2)$ ancilla qubits and $\tilde{O}(n^2/\epsilon^2)$ measurements. In contrast, we show that any protocol with zero ancilla, even a concatenating one, must make $\Omega(2^n/\epsilon^2)$ measurements, which is tight. Our results imply, to our knowledge, the first quantum learning task where logarithmically many qubits of quantum memory suffice for an exponential statistical advantage.

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.

The rapid advancement of quantum computing has led to an extensive demand for effective techniques to extract classical information from quantum systems, particularly in fields like quantum machine learning and quantum chemistry. However, quantum systems are inherently susceptible to noises, which adversely corrupt the information encoded in quantum systems. In this work, we introduce an efficient algorithm that can recover information from quantum states under Pauli noise. The core idea is to learn the necessary information of the unknown Pauli channel by post-processing the classical shadows of the channel. For a local and bounded-degree observable, only partial knowledge of the channel is required rather than its complete classical description to recover the ideal information, resulting in a polynomial-time algorithm. This contrasts with conventional methods such as probabilistic error cancellation, which requires the full information of the channel and exhibits exponential scaling with the number of qubits. We also prove that this scalable method is optimal on the sample complexity and generalise the algorithm to the weight contracting channel. Furthermore, we demonstrate the validity of the algorithm on the 1D anisotropic Heisenberg-type model via numerical simulations. As a notable application, our method can be severed as a sample-efficient error mitigation scheme for Clifford circuits.

Overparametrization is one of the most surprising and notorious phenomena in machine learning. Recently, there have been several efforts to study if, and how, Quantum Neural Networks (QNNs) acting in the absence of hardware noise can be overparametrized. In particular, it has been proposed that a QNN can be defined as overparametrized if it has enough parameters to explore all available directions in state space. That is, if the rank of the Quantum Fisher Information Matrix (QFIM) for the QNN's output state is saturated. Here, we explore how the presence of noise affects the overparametrization phenomenon. Our results show that noise can "turn on" previously-zero eigenvalues of the QFIM. This enables the parametrized state to explore directions that were otherwise inaccessible, thus potentially turning an overparametrized QNN into an underparametrized one. For small noise levels, the QNN is quasi-overparametrized, as large eigenvalues coexists with small ones. Then, we prove that as the magnitude of noise increases all the eigenvalues of the QFIM become exponentially suppressed, indicating that the state becomes insensitive to any change in the parameters. As such, there is a pull-and-tug effect where noise can enable new directions, but also suppress the sensitivity to parameter updates. Finally, our results imply that current QNN capacity measures are ill-defined when hardware noise is present.

Understanding the noise affecting a quantum device is of fundamental importance for scaling quantum technologies. A particularly important class of noise models is that of Pauli channels, as randomized compiling techniques can effectively bring any quantum channel to this form and are significantly more structured than general quantum channels. In this paper, we show fundamental lower bounds on the sample complexity for learning Pauli channels in diamond norm with unentangled measurements. We consider both adaptive and non-adaptive strategies. In the non-adaptive setting, we show a lower bound of $\Omega(2^{3n}\epsilon^{-2})$ to learn an $n$-qubit Pauli channel. In particular, this shows that the recently introduced learning procedure by Flammia and Wallman is essentially optimal. In the adaptive setting, we show a lower bound of $\Omega(2^{2.5n}\epsilon^{-2})$ for $\epsilon=\mathcal{O}(2^{-n})$, and a lower bound of $\Omega(2^{2n}\epsilon^{-2} )$ for any $\epsilon > 0$. This last lower bound even applies for arbitrarily many sequential uses of the channel, as long as they are only interspersed with other unital operations.