Dynamics of disordered, quantum many-body systems provide a classic physical setting for the breakdown of quantum ergodicity, known as many-body localization (MBL). In this novel out of equilibrium quantum phase, the many-body system acts as a local quantum memory along with preserving quantum-coherent macroscopic behavior even in highly excited states. I will review the properties of the MBL phase in one dimension and its efficient simulation using shallow quantum circuits (QC).
I will discuss the generalization of the shallow QC to explore MBL behavior in two dimensions and provide eigenstate signatures of the putative transition into the thermal phase. Furthermore, I will present instability arguments based on coupling MBL systems to finite thermal baths, highlighting the sensitivity of the phenomenon in higher dimensions.