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Recent works have investigated the emergence of a new kind of random matrixbehaviour in unitary dynamics following a quantum quench. Starting from atime-evolved state, an ensemble of pure states supported on a small subsystemcan be generated by performing projective measurements on the remainder of thesystem, leading to a projected ensemble. In chaotic quantum systems it wasconjectured that such projected ensembles become indistinguishable from theuniform Haar-random ensemble and lead to a quantum state design. Exact resultswere recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)]for the kicked Ising model at the self-dual point. We provide an alternativeconstruction that can be extended to general chaotic dual-unitary circuits withsolvable initial states and measurements, highlighting the role of theunderlying dual-unitarity and further showing how dual-unitary circuit modelsexhibit both exact solvability and random matrix behaviour. Building on resultsfrom biunitary connections, we show how complex Hadamard matrices and unitaryerror bases both lead to solvable measurement schemes.

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