In this work, we initiate the study of Hamiltonian learning for positive temperature bosonic Gaussian states, the quantum generalization of the widely studied problem of learning Gaussian graphical models. We obtain efficient protocols, both in sample and computational complexity, for the task of inferring the parameters of their underlying quadratic Hamiltonian under the assumption of bounded temperature, squeezing, displacement and maximal degree of the interaction graph. Our protocol only requires heterodyne measurements, which are often experimentally feasible, and has a sample complexity that scales logarithmically with the number of modes. Furthermore, we show that it is possible to learn the underlying interaction graph in a similar setting and sample complexity. Taken together, our results put the status of the quantum Hamiltonian learning problem for continuous variable systems in a much more advanced state when compared to spins, where state-of-the-art results are either unavailable or quantitatively inferior to ours. In addition, we use our techniques to obtain the first results on learning Gaussian states in trace distance with a quadratic scaling in precision and polynomial in the number of modes, albeit imposing certain restrictions on the Gaussian states. Our main technical innovations are several continuity bounds for the covariance and Hamiltonian matrix of a Gaussian state, which are of independent interest, combined with what we call the local inversion technique. In essence, the local inversion technique allows us to reliably infer the Hamiltonian of a Gaussian state by only estimating in parallel submatrices of the covariance matrix whose size scales with the desired precision, but not the number of modes. This way we bypass the need to obtain precise global estimates of the covariance matrix, controlling the sample complexity.

Variational quantum circuits (VQCs) [1] are candidates for achieving practical quantum advantages in the noisy intermediate-scale quantum (NISQ) era [2], when scalable error-corrected quantum computers are not yet available. VQCs utilize classical control to optimize a quantum circuit to solve computation problems, including optimization [3], eigen-system problem [4-10], partial-differential equations [11], quantum simulation [12-14] and machine learning [15-23]. As a general approach of designing quantum circuits, it has also found applications in the approximation [24], preparation [25, 26], classification [27-31] and tomography [32] of quantum states. Initial works on VQCs concern discrete-variable (DV) finite-dimensional systems of qubits, which are natural for computation; while continous-variable (CV) systems of bosonic qumodes are less explored. Yet, many important quantum systems are modelled by qumodes. For example, quantum communication and networking [33-37] rely on photons--the only flying quantum information carrier. In this regard, quantum transduction and entanglement distillation are shown to be enhanced by CV VQCs [38]; Photonic quantum computers [39, 40] are also relying on bosonic encoding such as the cat code and Gottesman-Kitaev-Preskill (GKP) code [41], which has shown great promise [42, 43]. The engineering of such code states are greatly boosted by CV VQCs [44-47]; Finally, distributed entangled sensor networks ubiquitously rely on CV VQCs to achieve quantum advantages in sensing [48-51] and data classification [52, 53]. Different from traditional algorithms, the runtime of VQCs is characterized by the time necessary to train the variational parameters to optimize a cost function.

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.

We miss general methods for quantum solitons, although they can act as entanglement generators or as self-organized quantum processors. We develop a computational approach that uses a neural network as a variational ansatz for quantum solitons in an array of waveguides. By training the resulting phase space quantum machine learning model, we find different soliton solutions varying the number of particles and interaction strength. We consider Gaussian states that enable measuring the degree of entanglement and sampling the probability distribution of many-particle events. We also determine the probability of generating particle pairs and unveil that soliton bound states emit correlated pairs. These results may have a role in boson sampling with nonlinear systems and in quantum processors for entangled nonlinear waves. A soliton is a non-perturbative solution of a classical nonlinear wave-equation; it may describe mean-field states of atoms (as in Bose-Einstein condensation) or photons (as in nonlinear optics) [1]. From a quantum mechanical perspective, a soliton may correspond to a coherent state; however, the nonlinearity may induce squeezing or non-Gaussianity [2]. The quantum properties of solitons inspired experimental investigations, as quantum non-demolition, squeezing [3-6] and photon bound states [7]. Authors reported on theoretical studies on the soliton quantum features, as evaporation and breathing [8-13].