Neural networks possess formidable representational power, rendering them invaluable in solving complex quantum many-body systems. While they excel at analyzing static solutions, nonequilibrium processes, including critical dynamics during a quantum phase transition, pose a greater challenge for neural networks. To address this, we utilize neural networks and machine learning algorithms to investigate the time evolutions, universal statistics, and correlations of topological defects in a one-dimensional transverse-field quantum Ising model. Specifically, our analysis involves computing the energy of the system during a quantum phase transition following a linear quench of the transverse magnetic field strength. The excitation energies satisfy a power-law relation to the quench rate, indicating a proportional relationship between the excitation energy and the kink numbers. Moreover, we establish a universal power-law relationship between the first three cumulants of the kink numbers and the quench rate, indicating a binomial distribution of the kinks. Finally, the normalized kink-kink correlations are also investigated and it is found that the numerical values are consistent with the analytic formula.

Abstract: Graph clustering is the process of grouping vertices into densely connected sets called clusters. In doing so, we obtain a heuristic approximation to the intra-cluster density maximization problem. We use two variations of a Boltzmann machine heuristic to obtain numerical solutions. For benchmarking purposes, we compare solution quality and computational performances to those obtained using a commercial solver, Gurobi. We also compare clustering quality to the clusters obtained using the popular Louvain modularity maximization method.

A central subject in Condensed Matter Physics and Statistical Mechanics is the study of phase transitions and critical phenomena [1, 2]. In the last decades, due to the increasing computer power resources, numerical methods have become an indispensable tool for the analysis of classical and quantum interacting systems. Most of these methods, such as Monte Carlo simulations, are performed at finite size systems, which demand the analysis by scaling theories to avoid misleading finite size effects [3-6]. However, depending on the type of systems (classical or quantum), or the geometry/dimensionality, performing a finite size scaling (FSS) analysis may be a challenge - sometimes an unfeasible task -, due to technical bottlenecks: for instance, as a paradigm, in quantum Monte Carlo simulations the occurrence of the infamous minus-sign problem, i.e. the occurrence a negative statistical weight, restricts the simulations to small lattice sizes [7-9]. Another instance is the analysis of three-dimensional systems, in which an extrapolation to the thermodynamic limit is very demanding, even in absence of the sign problem. In view of this, it is worth developing techniques that could give hints of the existing phases and their phase transitions at finite small system sizes, but, at the same time, could also provide quantitatively reasonable critical points. With the advent of big data analysis, e.g. with machine learning techniques, a great expectation is placed to this end. Indeed, over the past few years, there has been an effort to develop and benchmark supervised and unsupervised machine learning techniques [10-12].

Experimental quantum simulators have become large and complex enough that discovering new physics from the huge amount of measurement data can be quite challenging, especially when little theoretical understanding of the simulated model is available. Unsupervised machine learning methods are particularly promising in overcoming this challenge. For the specific task of learning quantum phase transitions, unsupervised machine learning methods have primarily been developed for phase transitions characterized by simple order parameters, typically linear in the measured observables. However, such methods often fail for more complicated phase transitions, such as those involving incommensurate phases, valence-bond solids, topological order, and many-body localization. We show that the diffusion map method, which performs nonlinear dimensionality reduction and spectral clustering of the measurement data, has significant potential for learning such complex phase transitions unsupervised. This method works for measurements of local observables in a single basis and is thus readily applicable to many experimental quantum simulators as a versatile tool for learning various quantum phases and phase transitions.