One qubit as a universal approximant

A single-qubit circuit can approximate any bounded complex function stored in the degrees of freedom defining its quantum gates. The single-qubit approximant presented in this work is operated through a series of gates that take as their parametrization the independent variable of the target function and an additional set of adjustable parameters. The independent variable is re-uploaded in every gate while the parameters are optimized for each target function. The output state of this quantum circuit becomes more accurate as the number of re-uploadings of the independent variable increases, i.e., as more layers of gates parameterized with the independent variable are applied. In this work, we provide two proofs of this claim related to both the Fourier series and the universal approximation theorem for neural networks, and we benchmark both methods against their classical counterparts. We further implement a single-qubit approximant in a real superconducting qubit device, demonstrating how the ability to describe a set of functions improves with the depth of the quantum circuit. This work shows the robustness of the re-uploading technique on quantum machine learning.We acknowledge financial support from Secretaria d’Universitatsi Recerca del Departament d’Empresa i Coneixement de la Generalitat de Catalunya, co-funded by the European Union Regional Development Fund within the ERDF Operational Program of Catalunya (project QuantumCat, ref. 001-P-001644). A.G-S received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 951911 (AI4Media). P. F.-D. acknowledges support from ”la Caixa” Foundation - Junior leader fellowship (ID100010434-CF/BQ/PR19/11700009), Ministry of Economy and Competitiveness and Agencia Estatal de Investigación (FIS2017-89860-P; SEV-2016-0588; PCI2019-111838-2), and European Commission (FET-Open AVaQus GA 899561; QuantERA). IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.Peer ReviewedPostprint (author's final draft