Jonas Landman

We introduce a Convolutional and a measurement based Pooling layer that offer polynomial advantages over their classical analogs. By conserving the subspace preserving structure of the state during the computation, these layers can be assembled to perform complex deep-learning algorithms such as Convolutional Neural Network architectures, while assuring the correct training of the quantum circuit. In particular, those circuits can avoid Barren Plateau by only considering subspaces of polynomial size, limiting the potential running time advantages to polynomial ones. Recent work has pointed out the link between the absence of Barren Plateau and a non-exponential advantage in the near-term QML literature, and we believe that our proposal offers a promising path for useful QML algorithms by optimizing the framework that avoids vanishing gradient phenomena. Our works also deal with an important question that only a few works address due to hardware limitations: how to ensure that a method's performance will scale with the size of the problems? By offering software tools that are tailored for Hamming-Weight preserving algorithms, and by mimicking the behavior of state-of-the-art classical deep-learning layers, we offer a solution that performs well in comparison with classical methods while offering an interesting running time advantage. Our software, that can be accessed through, allowed us to train our model on 10-label classification tasks that are far more complex than usual binary classification tasks used to illustrate QML methods and are commonly used in the classical Machine Learning literature.

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.