Design boosters: from constant-time quantum chaos to infty-designs and beyond

We study a counterintuitive property of 'conditioning' on the result of measuring a subsystem of a quantum state: such conditioning can boost design quality, at the cost of increased system size. We work in the setting of deep thermalization from many-body physics: starting from a bipartite state on a global system (A,B) drawn from a k-design, we measure system B in the computational basis, keep the outcome and examine the state that remains in system A, approximating the overall ensemble (the 'projected ensemble') by a k'-design. We ask: how does the design quality change due to this procedure, or how does k' compare to k? We give the first rigorous example of unitary dynamics generating a state such that, projection at very early (constant) times can boost design randomness. These dynamics are those of quantum chaos, modeled by the evolution of a Hamiltonian drawn from the Gaussian Unitary Ensemble (GUE). We show that, even though a state generated by such dynamics at constant time only forms a k=mathcal{O}(1) design, the projected ensemble is Haar-random or a $k'=infty$ design in the thermodynamic limit i.e. when $N_B=infty$. This phenomenon persists even with weaker and more physically realistic assumptions; our results can be appropriately applied to non-GUE Hamiltonians that nevertheless show likely chaotic signatures in their eigenbases. Moreover, we show that with no assumption on how the global state was generated, a k-design experiences a degradation in design quality to k' = lfloor k/2 rfloor. This improves upon best prior results on the deep thermalization of designs. Together, our contributions argue for design boosting as a result of chaos and showcase a novel mechanism to generate good designs.