Classifying phases of matter is key to our understanding of many problems in physics. For quantum-mechanical systems in particular, the task can be daunting due to the exponentially large Hilbert space. With modern computing power and access to ever-larger data sets, classification problems are now routinely solved using machine-learning techniques 1 . Here, we propose a neural-network approach to finding phase transitions, based on the performance of a neural network after it is trained with data that are deliberately labelled incorrectly. We demonstrate the success of this method on the topological phase transition in the Kitaev chain 2 , the thermal phase transition in the classical Ising model 3 , and the many-body-localization transition in a disordered quantum spin chain 4 . Our method does not depend on order parameters, knowledge of the topological content of the phases, or any other specifics of the transition at hand. It therefore paves the way to the development of a generic tool for identifying unexplored phase transitions.Machine learning as a tool for analysing data is becoming more and more prevalent in an increasing number of fields. This is due to a combination of availability of large amounts of data and the advances in hardware and computational power, the latter most notably through the use of graphical processing units.Two typical methods of machine learning can be distinguished, namely the unsupervised and supervised methods. In the former the machine receives no input other than the data and is asked, for example, to extract features or to cluster the samples. Such an unsupervised approach was applied to identify phase transitions and order parameters from images of classical configurations of Ising models 5 . In the supervised learning methods, the data have to be supplemented by a set of labels. A typical example is classification of data, where each sample is assigned a class label. The machine is trained to recognize samples and predict their associated label, demonstrating that it has learned by generalizing to samples it has not encountered before. This approach, too, has been demonstrated on Ising models Motivated by previous studies, we apply machine-learning techniques to the detection of phase transitions. In contrast to the earlier works, however, we focus on a combination of supervised and unsupervised techniques. In most cases, namely, it is exactly the labelling that one would like to find out (that is, classification of phases). That implies that a labelling is not known beforehand, and hence supervised techniques are not directly applicable. In this Letter we demonstrate that it is possible to find the correct labels, by purposefully mislabelling the data and evaluating the performance of the machine learner. We will base our method on NNs, which are capable of fitting arbitrary nonlinear functions 11 . Indeed, if a linear feature extraction method worked, there would have been no need to explicitly find labels in the first place.We emphasize the main result in this work is...

In this work we demonstrate that non-random mechanisms that lead to single-particle localization may also lead to many-body localization, even in the absence of disorder. In particular, we consider interacting spins and fermions in the presence of a linear potential. In the non-interacting limit, these models show the well known Wannier-Stark localization. We analyze the fate of this localization in the presence of interactions. Remarkably, we find that beyond a critical value of the potential gradient, these models exhibit non-ergodic behavior as indicated by their spectral and dynamical properties. These models, therefore, constitute a new class of generic non-random models that fail to thermalize. As such, they suggest new directions for experimentally exploring and understanding the phenomena of many-body localization. We supplement our work by showing that by employing machine learning techniques, the level statistics of a system may be calculated without generating and diagonalizing the Hamiltonian, which allows a generation of large statistics.

We study a one-dimensional model of interacting bosons on a lattice with two flat bands. Regular condensation is suppressed due to the absence of a well defined minimum in the single particle spectrum. We find that interactions stabilize a number of non-trivial phases like a pair (quasi-) condensate, a supersolid at incommensurable fillings and valence bond crystals at commensurability. We support our analytical calculations with numerical simulations using the density matrix renormalization group technique. Implications for cold-atoms and extensions to higher dimensions are discussed.

We demonstrate quantum many-body state reconstruction from experimental data generated by a programmable quantum simulator, by means of a neural network model incorporating known experimental errors. Specifically, we extract restricted Boltzmann machine (RBM) wavefunctions from data produced by a Rydberg quantum simulator with eight and nine atoms in a single measurement basis, and apply a novel regularization technique to mitigate the effects of measurement errors in the training data. Reconstructions of modest complexity are able to capture one-and two-body observables not accessible to experimentalists, as well as more sophisticated observables such as the Rényi mutual information. Our results open the door to integration of machine learning architectures with intermediate-scale quantum hardware. arXiv:1904.08441v2 [quant-ph]