Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuits
It provides a theoretical foundation for designing practical quantum neural networks on near-term devices, addressing a crucial gap in quantum machine learning, though it is incremental in advancing existing approximation theories.
This paper tackles the problem of understanding the expressive power of parameterized quantum circuits (PQCs) for approximating multivariate functions, establishing the first non-asymptotic error bounds in terms of qubits, depth, and parameters, and showing that for smooth functions, PQCs can be smaller than deep ReLU neural networks.
Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks. We further validate the approximation capability of PQCs through numerical experiments. Our results provide a theoretical foundation for designing practical PQCs and quantum neural networks for machine learning tasks that can be implemented on near-term quantum devices, paving the way for the advancement of quantum machine learning.