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.
Highlights
Recent years have witnessed a surge of interest in the question of whether and how quantum computers can meaningfully address computational problems in machine learning [1, 2]
We present a detailed discussion of the approach of Ref. [33], which demonstrates how the functions implemented by a parametrized quantum circuits (PQCs)-based model can be represented by generalized trigonometric polynomials
We have derived Rademacher complexity and metric entropy bounds for PQC-based model classes
Summary
Recent years have witnessed a surge of interest in the question of whether and how quantum computers can meaningfully address computational problems in machine learning [1, 2]. This development has been largely driven by two factors. The increasing availability of quantum computational devices provides significant stimulus While these “noisy intermediate-scale quantum” (NISQ) devices are still a far cry from full-scale fault-tolerant quantum computers, there exists growing evidence that they may be able to out-perform classical computers on some highly-tailored tasks [8]. Of particular prominence are variational quantum algorithms in which a parametrized quantum circuit (PQC) is used to define a machine learning model which is updated via a classical optimizer [10,11,12]
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