##### Encoding-dependent generalization bounds for parametrized quantum circuits

M.C. Caro, E. Gil-Fuster, J.J. Meyer, J. Eisert, R. Sweke

Quantum 5, 582 (2021).

A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.

##### Infinite-Dimensional Programmable Quantum Processors

M. Gschwendtner, A. Winter

PRX Quantum 2, 030308 (2021).

A universal programmable quantum processor uses "program" quantum states to apply an arbitrary quantum channel to an input state. We generalize the concept of a finite-dimensional programmable quantum processor to infinite dimension assuming an energy constraint on the input and output of the target quantum channels. By proving reductions to and from finite-dimensional processors, we obtain upper and lower bounds on the program dimension required to approximately implement energy-limited quantum channels. In particular, we consider the implementation of Gaussian channels. Due to their practical relevance, we investigate the resource requirements for gauge-covariant Gaussian channels. Additionally, we give upper and lower bounds on the program dimension of a processor implementing all Gaussian unitary channels. These lower bounds rely on a direct information-theoretic argument, based on the generalization from finite to infinite dimension of a certain "replication lemma" for unitaries.

##### Binary classification with classical instances and quantum labels

M.C. Caro

Quantum Machine Intelligence 3, 18 (2021).

In classical statistical learning theory, one of the most well-studied problems is that of binary classification. The information-theoretic sample complexity of this task is tightly characterized by the Vapnik-Chervonenkis (VC) dimension. A quantum analog of this task, with training data given as a quantum state has also been intensely studied and is now known to have the same sample complexity as its classical counterpart. We propose a novel quantum version of the classical binary classification task by considering maps with classical input and quantum output and corresponding classical-quantum training data. We discuss learning strategies for the agnostic and for the realizable case and study their performance to obtain sample complexity upper bounds. Moreover, we provide sample complexity lower bounds which show that our upper bounds are essentially tight for pure output states. In particular, we see that the sample complexity is the same as in the classical binary classification task w.r.t. its dependence on accuracy, confidence and the VC-dimension.

##### Necessary criteria for Markovian divisibility of linear maps

M.C. Caro, B. Graswald

Journal of Mathematical Physics 62, 042203 (2021).

We describe how to extend the notion of infinitesimal Markovian divisibility from quantum channels to general linear maps and compact and convex sets of generators. We give a general approach toward proving necessary criteria for (infinitesimal) Markovian divisibility. With it, we prove two necessary criteria for infinitesimal divisibility of quantum channels in any finite dimension d: an upper bound on the determinant in terms of a Theta (d)-power of the smallest singular value and in terms of a product of Theta (d) smallest singular values. These allow us to analytically construct, in any given dimension, a set of channels that contains provably non-infinitesimal Markovian divisible ones. Moreover, we show that, in general, no such non-trivial criteria can be derived for the classical counterpart of this scenario.

##### A stable quantum Darmois-Skitovich theorem

J. Cuesta

Journal of Mathematical Physics 61 (2), 022201 (2020).

The Darmois-Skitovich theorem is a simple characterization of the normal distribution in terms of the independence of linear forms. We present here a non-commutative version of this theorem in the context of Gaussian bosonic states and show that this theorem is strongly stable under small errors in its underlying conditions. An explicit estimate of the stability constants which depend on the physical parameters of the problem is given.

##### Are almost-symmetries almost linear?

J. Cuesta, M.M. Wolf

Journal of Mathematical Physics 60 (8), 082101 (2019).

It d-pends. Wigner's symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive epsilon in a separable Hilbert space admits a weak linear approximation, i.e., one relative to any fixed observable. This implies the existence of a linear approximation that is 4 epsilon d-close in Hilbert-Schmidt norm, with d the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of d does not exist in general. To this end, we provide a lower bound that depends logarithmically on d.

##### Quantum Zeno effect generalized

T. Möbus, M. Wolf.

Journal of Mathematical Physics 60, 052201 (2019).

The quantum Zeno effect, in its original form, uses frequent projective measurements to freeze the evolution of a quantum system that is initially governed by a fixed Hamiltonian. We generalize this effect simultaneously in three directions by allowing open system dynamics, time-dependent evolution equations and general quantum operations in place of projective measurements. More precisely, we study Markovian master equations with bounded generators whose time dependence is Lipschitz continuous. Under a spectral gap condition on the quantum operation, we show how frequent measurements again freeze the evolution outside an invariant subspace. Inside this space, the evolution is described by a modified master equation.