Encoding-dependent generalization bounds for parametrized quantum circuits

Matthias C. Caro1,2, Elies Gil-Fuster3,4, Johannes Jakob Meyer3, Jens Eisert3,4,5, and Ryan Sweke3

1Department of Mathematics, Technical University of Munich, 85748 Garching, Germany
2Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
3Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
4Fraunhofer Heinrich Hertz Institute, 10587 Berlin, Germany
5Helmholtz-Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany

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Updated version: The authors have uploaded version v3 of this work to the arXiv which may contain updates or corrections not contained in the published version v2. The authors left the following comment on the arXiv:
35 pages, 3 figures; corrected a mistake in Eq. (38) of the previous version, results remain unchanged except for a restriction to frequency vectors with integer entries; added a conjecture to recover results in full

Abstract

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.

In a typical supervised machine learning problem, one is given a fixed-size dataset consisting of labelled objects. For example, the dataset might contain images labelled as dogs or cats. The goal is then to use the dataset to learn some classification algorithm, which when given appropriate new objects – ones not already present in the dataset but of the same type – can label them correctly. Ideally, once one has obtained a classification algorithm, one would like to be able to have some sort of guarantee on how well this classification algorithm will perform – i.e., on how accurate it will be on previously unseen objects. Such guarantees on the “out-of-sample" performance of classification algorithms – i.e., on their accuracy on previously unseen data – are known as generalization bounds. Over time though it has been realized that various notions of complexity are critical for determining how well a class of classification algorithms will generalize. Intuitively, the more complex a classification algorithm can be, the more likely it is to "overfit" the available data, and therefore generalize badly. Indeed, we now know that one can derive a variety of generalization bounds for classification algorithms, which allow us to place guarantees on the "out-of-sample" performance of classification algorithms, in terms of their performance on the available data, and the complexity of the class of algorithms that they belong to.

Given the emerging availability of "noisy intermediate scale quantum" devices, a variety of approaches have been proposed for using such devices in supervised learning. One particularly important way in which these approaches differ, is in the way the classical data is entered into the quantum classification algorithm. In light of this, and given the importance of generalization bounds, it is natural to ask whether one can determine how the appropriate notions of complexity for generalization depend on the different strategies for encoding classical data into the quantum classifier.

In this work, we answer this question by quantifying rigorously how various notions of complexity depend on the strategy for encoding data into a quantum classifier. These results immediately allow us to state generalization bounds for such quantum classification algorithms. With these, we can give concrete recommendations for choosing classical-to-quantum encoding strategies.

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