Randomness boosters: From constant-time chaos to infinity-designs, and beyond

Many universal characteristics of chaotic quantum many-body systems can be understood through randomness. Here we provide strong evidence for an emergent random matrix universality using quantum state designs in the context of deep thermalization. Starting with a Hamiltonian modeled by the GUE, we identify times when even if the Hamiltonian evolution of the global state itself is only an O(1) design, the projected ensemble is exactly Haar (an infinity-design) in the thermodynamic limit. We discuss how our results can be appropriately applied to non-GUE Hamiltonians that nevertheless show chaotic signatures in their eigenbases. Thus, this first result constitutes a form of randomness boosting via chaos, which is a novel mechanism in the context of designs. Secondly, even without such chaotic characteristics in the global ensemble, we are able to go from a k to k/2 design via projection, significantly improving the previous best known result for projected ensembles by eliminating a bath size-dependence in the design quality.