Slimane Thabet

Quantum Machine Learning Algorithms based on Variational Quantum Circuits (VQCs) are important candidates for useful application of quantum computing. It is known that VQCs are linear models in some feature space of finite dimension. At first sight, if this feature map can be explicitly computed classically, one may wonder what is the interest of searching the best parameters of the quantum circuit instead of performing classically a linear regression on the same feature map, using a so called classical surrogate model schreiber2023classical. Even when the feature space is too large to be computed classically, methods exist to reduce its dimension by random sampling, realizing approximated classical models. At the same time, quantum advantages for learning tasks have been proven in the case of discrete data distributions and cryptography primitives. In this work, we present necessary conditions for a quantum model to avoid such dequantization, and we highlight that this could be only satisfied for high dimensional feature maps. Our study can be applied to any quantum circuits with continuous or discrete inputs, and propose conditions that guarantee a quantum model to remain far from its equivalent classical model. We show that this theory is compatible with previously proven quantum advantages on discrete inputs, and provides examples of advantages for continuous inputs. This separation is connected to large weight vector norm, and we suggest that this can only happen with a high dimensional feature map. Our results demonstrate that it is possible to design quantum models that cannot be classically approximated with good generalization. Finally, we discuss how concentration issues must be considered to design such instances. We expect that our work will be a starting point to design near-term quantum models that avoid dequantization methods by ensuring non-classical convergence properties, and to identify existing quantum models that can be classically approximated.