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Parameterized quantum circuits (PQCs) have emerged as a promising approachfor quantum neural networks. However, understanding their expressive power inaccomplishing machine learning tasks remains a crucial question. This paperinvestigates the expressivity of PQCs for approximating general multivariatefunction classes. Unlike previous Universal Approximation Theorems for PQCs,which are either nonconstructive or rely on parameterized classical dataprocessing, we explicitly construct data re-uploading PQCs for approximatingmultivariate polynomials and smooth functions. We establish the firstnon-asymptotic approximation error bounds for these functions in terms of thenumber of qubits, quantum circuit depth, and number of trainable parameters.Notably, we demonstrate that for approximating functions that satisfy specificsmoothness criteria, the quantum circuit size and number of trainableparameters of our proposed PQCs can be smaller than those of deep ReLU neuralnetworks. We further validate the approximation capability of PQCs throughnumerical experiments. Our results provide a theoretical foundation fordesigning practical PQCs and quantum neural networks for machine learning tasksthat can be implemented on near-term quantum devices, paving the way for theadvancement of quantum machine learning.

Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits