Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics

Recent works have investigated the emergence of a new kind random matrix behaviour in unitary dynamics following quantum quench. Starting from time-evolved state, an ensemble pure states supported on small subsystem can be generated by performing projective measurements remainder system, leading to projected ensemble. In chaotic systems it was conjectured that such projected ensembles become indistinguishable uniform Haar-random and lead mathvariant="italic">quantum state design. Exact results were recently presented Ho Choi [Phys. Rev. Lett. 128, 060601 (2022)] for kicked Ising model at self-dual point. We provide alternative construction extended general dual-unitary circuits with solvable initial measurements, highlighting role underlying dual-unitarity further showing how circuit models exhibit both exact solvability behaviour. Building biunitary connections, we show complex Hadamard matrices error bases measurement schemes.