  Recent works have investigated the emergence of a new kind of random matrix
behaviour in unitary dynamics following a quantum quench. Starting from a
time-evolved state, an ensemble of pure states supported on a small subsystem
can be generated by performing projective measurements on the remainder of the
system, leading to a projected ensemble. In chaotic quantum systems it was
conjectured that such projected ensembles become indistinguishable from the
uniform Haar-random ensemble and lead to a quantum state design. Exact results
were 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 alternative
construction that can be extended to general chaotic dual-unitary circuits with
solvable initial states and measurements, highlighting the role of the
underlying dual-unitarity and further showing how dual-unitary circuit models
exhibit both exact solvability and random matrix behaviour. Building on results
from biunitary connections, we show how complex Hadamard matrices and unitary
error bases both lead to solvable measurement schemes.
