Berlin 2024 – scientific programme
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QI: Fachverband Quanteninformation
QI 9: Quantum Machine Learning and Classical Simulability
QI 9.3: Talk
Tuesday, March 19, 2024, 10:15–10:30, HFT-FT 101
Parametrized Quantum Circuits and their approximation capacities in the context of quantum machine learning — Alberto Manzano1, •David Dechant2,3, Jordi Tura2,3, and Vedran Dunjko2,3,4 — 1Department of Mathematics and CITIC, Universidade da Coruña, Campus de Elviña s/n, A Coruña, Spain — 2Applied Quantum Algorithms Leiden, The Netherlands — 3Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands — 4LIACS, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, Netherlands
Parametrized quantum circuits (PQC) are used in recent approaches to quantum machine learning to learn various types of data, with an underlying expectation that if the PQC is made sufficiently deep, and the data plentiful, the generalization error will vanish, and the model will capture the essential features of the distribution. While there exist results proving the approximability of square-integrable functions by PQCs under the L2 distance, the approximation for other function spaces and under other distances has been less explored. In this work we show that PQCs can approximate the space of continuous functions, p-integrable functions and the Hk Sobolev spaces under specific distances. Moreover, we develop generalization bounds that connect different function spaces and distances. These results provide a theoretical basis for different applications of PQCs, for example for solving differential equations. Furthermore, they provide us with new insight on how to design PQCs and loss functions which better suit the specific needs of the users.
Keywords: Quantum machine learning; Parametrized quantum circuits; Generalization bounds; Quantum computing; Variational quantum algorithms