The advent of quantum simulators and quantum processors has enabled probing non-equilibrium quantum matter, simultaneously raising fundamental questions about entanglement, complexity and control of many-body states. Theoretical studies revealed new regimes of non-thermalising quantum dynamics (including prethermalization and many-body localization), associated with the emergence of exact or approximate conservation laws. I will discuss the state of the art in studying non-equilibrium matter with quantum processors. As an example, I will describe a recent observation of robust edge modes protected by prethermalization in a chain of up to 46 superconducting qubits. Remarkably, despite external noise, the ``wave function” of the edge modes can be reconstructed precisely from measuring dynamics of multi-qubit observables, providing an example of an intricate many-body quantity accessible with noisy quantum processors. I will conclude with an outlook, highlighting non-equilibrium phenomena which may soon be within reach of experiments, but are hard to simulate by classical methods.