Structural risk minimization for quantum linear classifiers

Abstract

Quantum machine learning (QML) stands out as one of the typically highlighted candidates for quantum computing's near-term "killer application". In this context, QML models based on parameterized quantum circuits comprise a family of machine learning models that are well suited for implementations on near-term devices and that can potentially harness computational powers beyond what is efficiently achievable on a classical computer. However, how to best use these models -e.g., how to control their expressivity to best balance between training accuracy and generalization performance -is far from understood. In this paper we investigate capacity measures of two closely related QML models called explicit and implicit quantum linear classifiers (also called the quantum variational method and quantum kernel estimator) with the objective of identifying new ways to implement structural risk minimization -i.e., how to balance between training accuracy and generalization performance. In particular, we identify that the rank and Frobenius norm of the observables used in the QML model closely control the model's capacity. Additionally, we theoretically investigate the effect that these model parameters have on the training accuracy of the QML model. Specifically, we show that there exists datasets that require a high-rank observable for correct classification, and that there exists datasets that can only be classified with a given margin using an observable of at least a certain Frobenius norm. Our results provide new options for performing structural risk minimization for QML models.

Encoding-dependent generalization bounds for parametrized quantum circuits

“…Once again, the results of Ref. [32] differ from ours in that they are applicable only to the encoding-first case, and that they do not provide an explicit dependence on the data-encoding strategy. However, similarly the perspective we advocate in this work, the authors of Ref.…”

“…Given the fundamental role of generalization bounds, there has recently been a strong and steady stream of works contributing to the derivation of generalization bounds for PQC-based models [24][25][26][27][28][29][30][31][32]. However, as discussed in detail in Section 4, these prior works all differ from our results in a variety of ways.…”

“…However, similarly the perspective we advocate in this work, the authors of Ref. [32] stress the application of generalization bounds for model selection, via structural risk minimization.…”

“…Working from the perspective of kernel methods, Ref. [32] has recently investigated the complexity of encoding-first PQCbased models in terms of properties of the parametrized measurement which follows data-encoding. More specifically, they interpret the entire parametrized circuit following the data-encoding as a parametrized measurement, and provide bounds for the VC-dimension of the model class in terms of the rank of the parametrized observable, and for the fat-shattering dimension in terms of the Frobenius norm of the parametrized observable.…”

Expressivity of Variational Quantum Machine Learning on the Boolean Cube

“…Essentially, f θ (x) maps x to a real number by taking the expectation of H(θ) with respect to ρ(x). This model is also called a quantum neural network [24], and in the context of classification, it has been called the explicit linear quantum classifier [52] or variational quantum classifier [14]. Since the expectation of an observable is continuous valued, for classification, some post-processing of the output is required to map it to the finite set of possible classes.…”