Statistical descriptions of stochastic quantum dynamics

Random-matrix theory provides a versatile framework for describing complex quantum systems, focusing on universal features only constrained by dimensionality and symmetry properties. In particular, the spectral fluctuations of random matrices provide a highly applicable benchmark for quantum chaos and ergodic phases. This thesis focuses on the spectral properties of complex many-body quantum systems as they dynamically approach a chaotic, ergodic phase, where initially localised information becomes dispersed and scrambled over the system degrees of freedom. Measures of scrambling and chaos bounds are typically formulated in terms of dynamical correlations, such as those characterised by out-of-time-ordered correlators. We instead utilise spectral statistics, particularly the spectral form factor, which analyses correlations between the eigenvalues of a system, as a sensitive diagnostic tool to provide insights into the temporal evolution towards chaotic behaviour. We explore these features in three steps. First, we investigate this theme in the context of random quantum circuits, contrasting entanglement dynamics when unitary gates are drawn from each of the circular ensembles of Dyson’s Threefold Way. By combining exact analytical results for the minimal case of two qubits and numerical results for the full circuit dynamics, we find that the imposition of time-reversal symmetric gates reduces entanglement generation in the circuits. Next, we introduce a scaling theory for maximally efficient quantum-dynamical scrambling and formulate chaos bounds that we show to be saturated by Dyson’s Brownian motion. Finally, we show how exact analytical and asymptotic results can be obtained for a wide class of systems, for which the Brownian Sachdev-Ye-Kitaev model serves as a template. The results of this thesis lay the foundations for a deeper understanding of complex many-body quantum dynamics from a unified statistical perspective.