Time projected ensembles and quantum state designs in dual unitary circuits

In this talk we will introduce the time projected ensemble. Time projected ensembles define a distribution on quantum states evolved by a quantum circuit, where minimal randomness is introduced to the evolution. Considering a brick-work floquet quantum circuit, we apply a random one-site unitary on a single qubit at each time step. Leveraging previous results in the literature we show for a broad class of brick-work circuits, this ensemble of states deeply thermalizes to the Haar ensemble for all moments in the infinite time limit. We next use a mixture of analytical and numerical approaches to study the equilibration process, characterizing the relevant time scales for this process. In particular, we provide evidence that this protocol is optimal (fastest) if the brick-work circuit is made up of dual unitary matrices.