The recent advances in machine learning algorithms have boosted the application of these techniques to the field of condensed matter physics, in order e.g. to classify the phases of matter at equilibrium or to predict the real-time dynamics of a large class of physical models. Typically in these works, a machine learning algorithm is trained and tested on data coming from the same physical model. Here we demonstrate that unsupervised and supervised machine learning techniques are able to predict phases of a non-exactly solvable model when trained on data of a solvable model. In particular, we employ a training set made by single-particle correlation functions of a non-interacting quantum wire and by using principal component analysis, k-means clustering, t-distributed stochastic neighbor embedding and convolutional neural networks we reconstruct the phase diagram of an interacting superconductor. We show that both the principal component analysis and the convolutional neural networks trained on the data of the non-interacting model can identify the topological phases of the interacting model. Our findings indicate that non-trivial phases of matter emerging from the presence of interactions can be identified by means of unsupervised and supervised techniques applied to data of non-interacting systems.

The Quantum Approximate Optimization Algorithm (QAOA) adopts a hybrid quantum-classical approach to find approximate solutions to variational optimization problems. In fact, it relies on a classical subroutine to optimize the parameters of a quantum circuit. In this work we present a Bayesian optimization procedure to fulfil this optimization task, and we investigate its performance in comparison with other global optimizers. We show that our approach allows for a significant reduction in the number of calls to the quantum circuit, which is typically the most expensive part of the QAOA. We demonstrate that our method works well also in the regime of slow circuit repetition rates, and that few measurements of the quantum ansatz would already suffice to achieve a good estimate of the energy. In addition, we study the performance of our method in the presence of noise at gate level, and we find that for low circuit depths it is robust against noise. Our results suggest that the method proposed here is a promising framework to leverage the hybrid nature of QAOA on the noisy intermediate-scale quantum devices.

The imputation of missing data is a common procedure in data analysis that consists in predicting missing values of incomplete data points. In this work we analyse a variational quantum circuit for the imputation of missing data. We construct variational quantum circuits with gates complexity $\mathcal {O}(N)$ and $\mathcal {O}(N^{2})$ that return the last missing bit of a binary string for a specific distribution. We train and test the performance of the algorithms on a series of datasets finding good convergence of the results. Finally, we test the circuit for generalization to unseen data. For simple systems, we are able to describe the circuit analytically, making possible to skip the tedious and unresolved problem of training the circuit with repetitive measurements. We find beforehand the optimal values of the parameters and we make use of them to construct an optimal circuit suited to the generation of truly random data.

The use of Neural Networks in quantum many-body theory has undergone a formidable rise in recent years. Among the many possible applications, their pattern recognition power can be utilized when dealing with the study of equilibrium phase diagrams. Learning by Confusion has emerged as an interesting and unbiased scheme within this context. This technique involves systematically reassigning labels to the data in various ways, followed by training and testing the Neural Network. While random labeling results in low accuracy, the method reveals a peak in accuracy when the data are correctly and meaningfully partitioned, even if the correct labeling is initially unknown. Here, we propose a generalization of this confusion scheme for systems with more than two phases, for which it was originally proposed. Our construction relies on the use of a slightly different Neural Network: from a binary classifier, we move to a ternary one, which is more suitable to detect systems exhibiting three phases. After introducing this construction, we test it on free and interacting Kitaev chains and on the one-dimensional Extended Hubbard model, consistently achieving results that are compatible with previous works. Our work opens the way to wider use of Learning by Confusion, demonstrating once more the usefulness of Machine Learning to address quantum many-body problems.

The recent advances in machine learning algorithms have boosted the application of these techniques to the field of condensed matter physics, in order e.g. to classify the phases of matter at equilibrium or to predict the real-time dynamics of a large class of physical models. Typically in these works, a machine learning algorithm is trained and tested on data coming from the same physical model. Here we demonstrate that unsupervised and supervised machine learning techniques are able to predict phases of a non-exactly solvable model when trained on data of a solvable model. In particular, we employ a training set made by single-particle correlation functions of a non-interacting quantum wire and by using principal component analysis, k-means clustering, and convolutional neural networks we reconstruct the phase diagram of an interacting superconductor. We show that both the principal component analysis and the convolutional neural networks trained on the data of the non-interacting model can identify the topological phases of the interacting model with a high degree of accuracy. Our findings indicate that non-trivial phases of matter emerging from the presence of interactions can be identified by means of unsupervised and supervised techniques applied to data of non-interacting systems.