Encoding-dependent generalization bounds for parametrized quantum circuits

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 general...

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Autores principales:	Matthias C. Caro, Elies Gil-Fuster, Johannes Jakob Meyer, Jens Eisert, Ryan Sweke
Formato:	article
Lenguaje:	EN
Publicado:	Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften 2021
Materias:	Physics QC1-999
Acceso en línea:	https://doaj.org/article/fb1662ccb5434e1ba0fb7ee8d65d5868
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spelling	oai:doaj.org-article:fb1662ccb5434e1ba0fb7ee8d65d58682021-11-17T12:17:37ZEncoding-dependent generalization bounds for parametrized quantum circuits2521-327X10.22331/q-2021-11-17-582https://doaj.org/article/fb1662ccb5434e1ba0fb7ee8d65d58682021-11-01T00:00:00Zhttps://quantum-journal.org/papers/q-2021-11-17-582/pdf/https://doaj.org/toc/2521-327XA 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.Matthias C. CaroElies Gil-FusterJohannes Jakob MeyerJens EisertRyan SwekeVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenarticlePhysicsQC1-999ENQuantum, Vol 5, p 582 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	Physics QC1-999
spellingShingle	Physics QC1-999 Matthias C. Caro Elies Gil-Fuster Johannes Jakob Meyer Jens Eisert Ryan Sweke Encoding-dependent generalization bounds for parametrized quantum circuits
description	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.
format	article
author	Matthias C. Caro Elies Gil-Fuster Johannes Jakob Meyer Jens Eisert Ryan Sweke
author_facet	Matthias C. Caro Elies Gil-Fuster Johannes Jakob Meyer Jens Eisert Ryan Sweke
author_sort	Matthias C. Caro
title	Encoding-dependent generalization bounds for parametrized quantum circuits
title_short	Encoding-dependent generalization bounds for parametrized quantum circuits
title_full	Encoding-dependent generalization bounds for parametrized quantum circuits
title_fullStr	Encoding-dependent generalization bounds for parametrized quantum circuits
title_full_unstemmed	Encoding-dependent generalization bounds for parametrized quantum circuits
title_sort	encoding-dependent generalization bounds for parametrized quantum circuits
publisher	Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
publishDate	2021
url	https://doaj.org/article/fb1662ccb5434e1ba0fb7ee8d65d5868
work_keys_str_mv	AT matthiasccaro encodingdependentgeneralizationboundsforparametrizedquantumcircuits AT eliesgilfuster encodingdependentgeneralizationboundsforparametrizedquantumcircuits AT johannesjakobmeyer encodingdependentgeneralizationboundsforparametrizedquantumcircuits AT jenseisert encodingdependentgeneralizationboundsforparametrizedquantumcircuits AT ryansweke encodingdependentgeneralizationboundsforparametrizedquantumcircuits
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Encoding-dependent generalization bounds for parametrized quantum circuits

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