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.
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.
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 mixed-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.
Quantum channels can be characterized by their action on an orthogonal operator basis, where these operators are related to observable properties of the quantum system. For qudit and multimode bosonic systems, this is encoded respectively in the Heisenberg--Weyl transfer matrix estimated from the Choi-state, and in the characteristic-function transfer map estimated from a two-mode squeezed vacuum based Choi-state. We derive sample-complexity bounds for estimating entries of the transfer matrix/map to additive accuracy ε with success probability ≥1−δ, under different resources: access to the complex-conjugate channel E∗ and/or parallel access to c copies. In all settings, the learner uses parallel channel calls with adaptively chosen, ancilla-assisted input states and measurements. Absolute values of transfer-matrix entries can be learned efficiently with simultaneous access to E and E∗, with tight scaling ε−4. Without conjugate access, any c<d copies are insufficient for efficient learning, requiring sample complexity exponential in the number of (d-level) qudits n (for prime d). Efficiency is recovered at c=d, with tight scaling ε−2d. For bosonic systems, exponential sample complexity holds in terms of an effective dimension induced by an energy constraint for all c=O(1/ε). Although the task is learning a particular state, these bounds carry stronger implications than standard state-learning bounds since the learner controls the inputs and has ancillary assistance. This establishes a hierarchy of channel-learning resources: self-complex-conjugate channels require two-copy ancilla-assisted access for efficient learning, while for every square-free d, some channels require d-copy access. As a corollary, we derive tighter lower bounds for state learning with limited multi-copy access.
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.
We consider the problem of distinguishing, with minimum probability of error, two optical beam-splitter channels with unequal complex-valued reflectivities using general quantum probe states entangled over M signal and ${M}^{\ensuremath{'}}$ idler mode pairs of which the signal modes are bounced off the beam splitter while the idler modes are retained losslessly. We obtain a lower bound on the output state fidelity valid for any pure input state. We define number-diagonal signal (NDS) states to be input states whose density operator in the signal modes is diagonal in the multimode number basis. For such input states, we derive series formulas for the optimal error probability, the output state fidelity, and the Chernoff-type upper bounds on the error probability. For the special cases of quantum reading of a classical digital memory and target detection (for which the reflectivities are real valued), we show that for a given input signal photon probability distribution, the fidelity is minimized by the NDS states with that distribution and that for a given average total signal energy Ns, the fidelity is minimized by any multimode Fock state with Ns total signal photons. For reading of an ideal memory, it is shown that Fock state inputs minimize the Chernoff bound. For target detection under high-loss conditions, a no-go result showing the lack of appreciable quantum advantage over coherent state transmitters is derived. A comparison of the error probability performance for quantum reading of number state and two-mode squeezed vacuum state (or EPR state) transmitters relative to coherent state transmitters is presented for various values of the reflectances. While the nonclassical states in general perform better than the coherent state, the quantitative performance gains differ depending on the values of the reflectances. The experimental outlook for realizing nonclassical gains from number state transmitters with current technology at moderate to high values of the reflectances is argued to be good.
Generalization is a central concept in machine learning theory, yet for quantum models, it is predominantly analyzed through uniform bounds that depend on a model's overall capacity rather than the specific function learned. These capacity-based uniform bounds are often too loose and entirely insensitive to the actual training and learning process. Previous theoretical guarantees have failed to provide non-uniform, data-dependent bounds that reflect the specific properties of the learned solution rather than the worst-case behavior of the entire hypothesis class. To address this limitation, we derive the first PAC-Bayesian generalization bounds for a broad class of quantum models by analyzing layered circuits composed of general quantum channels, which include dissipative operations such as mid-circuit measurements and feedforward. Through a channel perturbation analysis, we establish non-uniform bounds that depend on the norms of learned parameter matrices; we extend these results to symmetry-constrained equivariant quantum models; and we validate our theoretical framework with numerical experiments. This work provides actionable model design insights and establishes a foundational tool for a more nuanced understanding of generalization in quantum machine learning.
As quantum processors are deployed across different hardware platforms and remote cloud laboratories, a basic physical question is whether two black-box devices realize the same quantum process, without relying on a trusted classical description. We formulate the core primitive for this comparison task as \emph{distributed similarity estimation of quantum channels} (DSEC): given local access to two unknown channels, estimate the normalized inner product of their Choi states. We prove that the optimal query complexity of DSEC is Θ(max{d/ε,1/ε2}), where d is the channel dimension and ε is the additive error. This matching query complexity is nontrivial: channel learning permits input choices and interleaving known operations, which makes channel learning strictly harder than state learning. We first prove an information-theoretic lower bound with this scaling, which holds even in the \emph{strongest setting}, allowing adaptive strategies, multiple rounds of classical communication, and coherent access with arbitrary ancillas. We then give a matching upper bound in the \emph{weakest setting}, namely non-adaptive and ancilla-free incoherent access, via a randomized measurement algorithm achieving this bound. Finally, we show that our algorithm achieves a quadratic improvement over classical shadow baselines. Our results provide theoretically optimal and practical algorithms for quantum device benchmarking and distributed quantum learning.