In recent years both from fundamental and applied points of view, ergodicity-breaking phenomena start to play an important role in quantum memory applications, quantum algorithms, and deep machine learning. Among them, one should mention a many-body localization, integrability, topological phenomena. Many-body localization (MBL) provides a unique property to avoid thermalization and to keep information about the system’s initial state via the emergent local integrals of motion with increasing disorder. This opens new horizons in applying such systems in the area of quantum memory, machine learning and speeding up quantum algorithms. MBL being a generic phenomenon is not subject to fine-tuned nature attributed to the conventional integrable systems.
The complexity of above mentioned generic many-body problems prevents one from analytically rigorous description of it. Thus, it is of particular concern and high demand to model essential attributes of these phenomena in a universal manner. In this respect, the random-matrix setting provides the perfect and more tractable playground, which can be used as quantum simulations of many-body systems.
Even at smaller disorder values below MBL interacting systems might show anomalously slow thermalizing behavior characterized by multifractal states. Such states provide a unique opportunity for speeding-up quantum annealing or parallel tempering. Multifractal states are also useful for swift and accurate sampling of rare local minima in the Hilbert space crucially needed for efficient training in machine learning. In this part called "Non- ergodic phases of matter" the main purposes are to suggest and investigate disordered models with the robust non-ergodic phases of matter for their application to machine learning, quantum annealing, and quantum memory. In the related direction "Relations between entanglement and multifractality" we consider the role of entanglement and multifractality in quantum dynamics and thermalization of many- body disordered systems in MBL and thermalizing parameter range.
