Eliott Mamon

We study the dimension of the manifold of quantum states (called orbit) that a given quantum state of light can reach under the dynamics of linear or Gaussian optics. That is, we investigate how many directions in the Hilbert space a given state can explore under these sub-universal regimes. We find that these orbit dimensions reveal fundamental insights into the structure of attainable state spaces (e.g. boson bunching does not increase the number of accessible directions) with multi-faceted consequences. By showcasing a simple way to compute this topological quantity, we reveal how it can alone yield no-go results for some transformations. Our framework is provably applicable in both Fock representations and some phase-space representations like the Wigner and stellar representations. We study genericity and robustness properties of orbit dimensions, and propose strategies to probe them using homodyne/heterodyne measurements on pure states, or photon counters on two copies of general states. We also highlight how under Gaussian unitaries, orbit dimensions witness non-Gaussianity. Lastly, we rigorously establish implications of orbit dimensions for the number of directions that a bosonic variational circuit can explore in state space. Our approach sheds new light on the structure of reachable states in quantum optics, which can help practitioners understand limitations and sources of expressivity or non-Gaussianity in optical experiments.

Quantum Machine Learning (QML) has become a promising area for real world applications of quantum computers, and near-term methods and their scalability are still important research topics. A consequent amount of efforts has been put into understanding how to avoid Barren Plateaus (BPs), a vanishing gradient phenomenon that prevents the variational algorithms from being trained efficiently. In particular, evidence has recently been shown that the structures that allow us to avoid BP seem to allow classical simulation techniques. In addition, other important questions must be tackled to design near-term quantum algorithms that may offer an advantage. How to ensure that the performance of the algorithms will scale with input size, and how to compare classical and quantum algorithms on different figures of merit for a same use case? Recent works have proposed to use subspace preserving quantum circuits to mimic classical neural network architectures. By restricting the Hilbert space to a subspace of polynomial size with respect to the number of qubits, such architectures are likely to avoid BPs. This comes at the cost of that is, a classical method can perform the same computation in polynomial time. In this work, we propose a paradigm shift: focusing on subspace-preserving methods that aim for a practical polynomial advantage. In particular, we propose to use linear optical circuits that are intrinsically subspace preserving as they conserve the number of particles during the computation. We believe that this approach could be sufficient to create useful QML applications as the generation of Fock states with few particles can be extremely high. In this talk, we will present two recent contributions from our team published in Physical Review Research. [1] and Advanced Photonics [2]. First, we will recall how linear optical circuit are limited in their expressivity due to the photonic homomorphism described by Aaronson and Arkhipov. We propose in [1] a new scheme for near-term photonic quantum devices that allows to increase the expressive power of the quantum models beyond what linear optics can do. This scheme relies upon State Injection (SI), a measurement-based technique that can produce states that are more controllable, and solve learning tasks that are believed to be intractable classically. Then we will show how using [2] how we propose to adapt a subspace preserving Quantum Convolutional Neural Network (QCNN) architecture for linear optic setting with SI adaptivity. We realize a proof-of-concept experiment by employing a cutting-edge single-photon source based on a semiconductor Quantum Dot (QD) , a time-to-spatial demultiplexer, and universal programmable 12-mode and 8-mode interferometers realized with the femtosecond laser-writing technique. The designed PQCNN scheme is tailored to the experimental platform at hand, with the goal of carrying out a binary image classification. As a complement to the experimental investigation, we provide a systematic study on the scaling and complexity of the protocol, by leveraging numerical simulations on larger quantum systems, demonstrating the potential behind the proposed scheme for PQCNNs.

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.