How much can the action of an unknown unitary dynamics on one family of states reveal about its action elsewhere? We develop a general framework for out-of-distribution quantum dynamics learning based on the first and second moments of the input distribution. These define a notion of the bias and expressivity of an ensemble, which together control generalization between arbitrary pure-state training and testing distributions. Our framework shows that only studying the action of a unitary on random real states can learn its action on random complex states, that tensor products of unbiased pure-state ensembles preserve expressivity, and that single-qubit risks are independent of bias. It also recovers earlier locally scrambling results as loose special cases and yields tight product-to-Haar bounds. We conclude with a series of no-go results and limited generalization bounds for learning certain channels, which further serve to underscore the distinctive role of unitarity in enabling out-of-distribution generalization.
How much can the action of an unknown unitary dynamics on one family of states reveal about its action elsewhere? We develop a general framework for out-of-distribution quantum dynamics learning based on the first and second moments of the input distribution. These define a notion of the bias and expressivity of an ensemble, which together control generalization between arbitrary pure-state training and testing distributions. Our framework shows that only studying the action of a unitary on random real states can learn its action on random complex states, that tensor products of unbiased pure-state ensembles preserve expressivity, and that single-qubit risks are independent of bias. It also recovers earlier locally scrambling results as loose special cases and yields tight product-to-Haar bounds. We conclude with a series of no-go results and limited generalization bounds for learning certain channels, which further serve to underscore the distinctive role of unitarity in enabling out-of-distribution generalization.
Pure-state deep thermalization predicts that, after sufficient scrambling, projected ensembles approach universal distributions that contain information beyond the reduced density matrix. In realistic quantum systems, however, noise is unavoidable and the projected states are generally mixed. Whether mixed-state deep thermalization admits an analogous universal limiting distribution has remained unclear. Here we answer this question for quantum dynamics containing local quantum channels, i.e., errors. We show that the late-time mixed-state projected ensemble is completely determined by the normalized Choi state of the error channel and is universal, independent of the microscopic details of the underlying unitary dynamics. We establish this result analytically for dual-unitary circuits, global Haar-random dynamics, and local Haar-random circuits at large local dimension, and verify it numerically in qubit circuits and kicked Ising dynamics. In a solvable setting, we further find exponentially fast convergence to the limiting ensemble. Our results extend deep thermalization to noisy quantum dynamics and establish a direct correspondence between quantum errors and universal random mixed-state ensembles.
Generalization bounds are a critical tool to assess the training data requirements of Quantum Machine Learning (QML). Recent work has established guarantees for in-distribution generalization of quantum neural networks (QNNs), where training and testing data are drawn from the same data distribution. However, there are currently no results on out-of-distribution generalization in QML, where we require a trained model to perform well even on data drawn from a different distribution to the training distribution. Here, we prove out-of-distribution generalization for the task of learning an unknown unitary. In particular, we show that one can learn the action of a unitary on entangled states having trained only product states. Since product states can be prepared using only single-qubit gates, this advances the prospects of learning quantum dynamics on near term quantum hardware, and further opens up new methods for both the classical and quantum compilation of quantum circuits.
Approximate unitary designs reproduce the statistics of Haar-random unitaries to a given order and accuracy. Recent constructions realize such designs with logarithmic-depth circuits, enabling shallow measurement protocols for various quantum state-learning tasks while preserving performance. These results raise the question of whether approximate designs can replace exact designs more generally. We show that even exponentially small design error need not preserve the learning guarantees of exact designs. This motivates deriving learning guarantees directly from the measurement circuit structure. For observable estimation with classical shadows, we prove that logarithmic-depth two-layer Clifford circuits yield an unbiased estimator matching the global Clifford variance scaling for every state and Hermitian observable. Beyond observable estimation, this bound allows shallow measurements to retain global Clifford guarantees for other tasks, including setting-efficient tomography, mixed-state metrology, and stabilizer structure learning. We complement these statistical guarantees with a compact, exact tensor-network representation of the inverse shadow channel. When measurement bases are reused, approximate unitary designs of arbitrarily high order need not uniformly match the Haar variance in the many-shot limit. For the all-to-all random two-local circuits considered, doing so requires nearly linear depth. Together, our results reveal the capabilities and limitations of shallow quantum state learning, highlighting the distinction between approximating Haar randomness and reproducing its learning guarantees.
We investigate the problem of predicting the output behavior of unknown quantum channels. Given query access to an n-qubit channel E and an observable O, we aim to learn the mapping \begin{equation*} ρ\mapsto \mathrm{Tr}(O E[ρ]) \end{equation*} to within a small error for most ρ sampled from a distribution D. Previously, Huang, Chen, and Preskill proved a surprising result that even if E is arbitrary, this task can be solved in time roughly nO(log(1/ε)), where ε is the target prediction error. However, their guarantee applied only to input distributions D invariant under all single-qubit Clifford gates, and their algorithm fails for important cases such as general product distributions over product states ρ. In this work, we propose a new approach that achieves accurate prediction over essentially any product distribution D, provided it is not "classical" in which case there is a trivial exponential lower bound. Our method employs a "biased Pauli analysis," analogous to classical biased Fourier analysis. Implementing this approach requires overcoming several challenges unique to the quantum setting, including the lack of a basis with appropriate orthogonality properties. The techniques we develop to address these issues may have broader applications in quantum information.
When experimentally learning the action of a continuous-variable quantum process by probing it with inputs, there will often be some restriction on the input states used. One experimentally simple way to probe a quantum channel is to use low-energy coherent states. Learning a quantum channel in this way presents difficulties, due to the fact that two channels may act similarly on low-energy inputs but very differently for high-energy inputs. They may also act similarly on coherent-state inputs but differently on nonclassical inputs. Extrapolating the behavior of a channel for more general input states from its action on the far more limited set of low-energy coherent states is a case of out-of-distribution generalization. To be sure that such generalization gives meaningful results, one needs to relate error bounds for the training set to bounds that are valid for all inputs. We show that for any pair of channels that act sufficiently similarly on low-energy coherent-state inputs, one can bound how different the input-output relations are for any (high-energy or highly nonclassical) input. This proves that out-of-distribution generalization is always possible for learning quantum channels using low-energy coherent states, as long as enough samples are used.
Over-parameterization is commonly used to increase the expressivity of variational quantum circuits (VQCs), yet deeper and more highly parameterized circuits often exhibit poor trainability and limited generalization. In this work, we provide a theoretical explanation for this phenomenon from a function-class perspective. We show that sufficiently expressive, unstructured variational ansatze enter a Haar-like universality class in which both observable expectation values and parameter gradients concentrate exponentially with system size. As a consequence, the hypothesis class induced by such circuits collapses with high probability to a narrow family of near-constant functions, a phenomenon we term simplicity bias, with barren plateaus arising as a consequence rather than the root cause. Using tools from random matrix theory and concentration of measure, we rigorously characterize this universality class and establish uniform hypothesis-class collapse over finite datasets. We further show that this collapse is not unavoidable: tensor-structured VQCs, including tensor-network-based and tensor-hypernetwork parameterizations, lie outside the Haar-like universality class. By restricting the accessible unitary ensemble through bounded tensor rank or bond dimension, these architectures prevent concentration of measure, preserve output variability for local observables, and retain non-degenerate gradient signals even in over-parameterized regimes. Together, our results unify barren plateaus, expressivity limits, and generalization collapse under a single structural mechanism rooted in random-matrix universality, highlighting the central role of architectural inductive bias in variational quantum algorithms.