CMT Forum: Max McGinley

While the dynamics of many-body quantum systems can be extraordinarily complex, quantitative predictions can often be made by identifying appropriate statistical ensembles that capture the universal behaviour of a broad class of systems. In the context of dynamics, the late-time behaviour of scrambling systems without conservation laws are expected to be described by the spherical Haar ensemble—a uniform distribution over all states in Hilbert space. By understanding the key properties of Haar-random states, and the mechanisms by which they can emerge, we can understand the physics of a diverse range of systems in a unified way.

In this talk, I will describe new universal statistical theories that capture the dynamics of systems in regimes beyond Haar, focusing in particular on the states generated by early-time dynamics. These states, which are far from being thermalized, are much ‘simpler’ that Haar-random states, possessing short-range correlations and entanglement; nevertheless, their output distributions can still be complex, and classically hard to simulate. We argue that the behaviour of these early-time states can be captured by the so-called Scrooge ensemble [PRA 49, 668 (1994)], a more structured generalization of the Haar ensemble. As well as presenting evidence that the Scrooge ensemble describes the outputs of random constant-depth quantum circuits, I will show how our hypothesis accounts for the observed complexity of constant-time quantum dynamics, and predicts a dramatic susceptibility of these states to small amounts of noise. I will conclude by illustrating how the Scrooge ensemble could be used to describe other kinds of dynamics with beyond-Haar-random structure, such as the volume-law phase of monitored quantum circuits, and dynamics with conservation laws.