Quantum machine learning (QML) seeks to exploit the intrinsic properties of quantum mechanical systems, including superposition, coherence, and quantum entanglement for classical data processing. However, due to the exponential growth of the Hilbert space, QML faces practical limits in classical simulations with the state-vector representation of quantum system. On the other hand, phase-space methods offer an alternative by encoding quantum states as quasi-probability functions. Building on prior work in qubit phase-space and the Stratonovich-Weyl (SW) correspondence, we construct a closed, composable dynamical formalism for one- and many-qubit systems in phase-space. This formalism replaces the operator algebra of the Pauli group with function dynamics on symplectic manifolds, and recasts the curse of dimensionality in terms of harmonic support on a domain that scales linearly with the number of qubits. It opens a new route for QML based on variational modelling over phase-space.

Cubic-phase states are a sufficient resource for universal quantum computing over continuous variables. We present results from numerical experiments in which deep neural networks are trained via reinforcement learning to control a quantum optical circuit for generating cubic-phase states, with an average success rate of 96%. The only non-Gaussian resource required is photon-number-resolving measurements. We also show that the exact same resources enable the direct generation of a quartic-phase gate, with no need for a cubic gate decomposition.

The emergence of classical behavior from quantum mechanics as Planck's constant $\hbar$ approaches zero remains a fundamental challenge in physics [1-3]. This paper introduces a novel approach employing deep neural networks to directly learn the dynamical mapping from initial quantum state parameters (for Gaussian wave packets of the one-dimensional harmonic oscillator) and $\hbar$ to the parameters of the time-evolved Wigner function in phase space [4-6]. A comprehensive dataset of analytically derived time-evolved Wigner functions was generated, and a deep feedforward neural network with an enhanced architecture was successfully trained for this prediction task, achieving a final training loss of ~ 0.0390. The network demonstrates a significant and previously unrealized ability to accurately capture the underlying mapping of the Wigner function dynamics. This allows for a direct emulation of the quantum-classical transition by predicting the evolution of phase-space distributions as $\hbar$ is systematically varied. The implications of these findings for providing a new computational lens on the emergence of classicality are discussed, highlighting the potential of this direct phase-space learning approach for studying fundamental aspects of quantum mechanics. This work presents a significant advancement beyond previous efforts that focused on learning observable mappings [7], offering a direct route via the phase-space representation.

The task of testing whether two uncharacterized quantum devices behave in the same way is crucial for benchmarking near-term quantum computers and quantum simulators, but has so far remained open for continuous-variable quantum systems. In this Letter, we develop a machine learning algorithm for comparing unknown continuous variable states using limited and noisy data. The algorithm works on non-Gaussian quantum states for which similarity testing could not be achieved with previous techniques. Our approach is based on a convolutional neural network that assesses the similarity of quantum states based on a lower-dimensional state representation built from measurement data. The network can be trained offline with classically simulated data from a fiducial set of states sharing structural similarities with the states to be tested, or with experimental data generated by measurements on the fiducial states, or with a combination of simulated and experimental data. We test the performance of the model on noisy cat states and states generated by arbitrary selective number-dependent phase gates. Our network can also be applied to the problem of comparing continuous variable states across different experimental platforms, with different sets of achievable measurements, and to the problem of experimentally testing whether two states are equivalent up to Gaussian unitary transformations.

The last years have seen an incredible advancement in hardware solutions for quantum technologies. In particular, the recent demonstration of a quantum computational advantage via photonic circuits [1, 2] finally paves the way for the realization of full-fledged quantum information processing with light, a solution that bears intrinsic advantages with respect to other platforms, in terms of scalability, robustness and deployability [3, 4, 5]. At the same time, the increased control of infinite-dimensional quantum states in several other platforms, such as cavity [6, 7] or mechanical resonators [8], is pushing the boundaries of continuous-variable (CV) quantum information processing beyond photonics. Finally, the increased interplay between qubit and CV platforms [9, 10] spurs the interest into the development of quantum error correction codes [11, 12, 13] and provides an alternative to more standard approaches for quantum technologies. A combination of the aforementioned events thus marks a renewed surge of interest into CV information processing. From a theoretical perspective, the characterization of the information-processing capabilities of quantum devices has been recently subject to a paradigm shift, thanks to the introduction of statistical learning techniques [14, 15, 16, 17, 18], which underly the success of classical machine learning [19, 20, 21]. In this approach, one recognizes that a successful use of quantum devices often requires two ingredients: (i) the estimation of quantities of interest about the quantum states or processes running in the device; (ii) the optimization of the device's parameter setup based on the estimated data, in order to maximize the device's performance in a specific task.

Machine Learning has wide applications in a broad range of subjects, including physics. Recent works have shown that neural networks can learn classical Hamiltonian mechanics. The results of these works motivate the following question: Can we endow neural networks with inductive biases coming from quantum mechanics and provide insights for quantum phenomena? In this work, we try answering these questions by investigating possible approximations for reconstructing the Hamiltonian of a quantum system given one of its wave--functions. Instead of handcrafting the Hamiltonian and a solution of the Schr\"odinger equation, we design neural networks that aim to learn it directly from our observations. We show that our method, termed Quantum Potential Neural Networks (QPNN), can learn potentials in an unsupervised manner with remarkable accuracy for a wide range of quantum systems, such as the quantum harmonic oscillator, particle in a box perturbed by an external potential, hydrogen atom, P\"oschl--Teller potential, and a solitary wave system. Furthermore, in the case of a particle perturbed by an external force, we also learn the perturbed wave function in a joint end-to-end manner.