  Quantum chaos in many-body systems provides a bridge between statistical and
quantum physics with strong predictive power. This framework is valuable for
analyzing properties of complex quantum systems such as energy spectra and the
dynamics of thermalization. While contemporary methods in quantum chaos often
rely on random ensembles of quantum states and Hamiltonians, this is not
reflective of most real-world systems. In this paper, we introduce a new
perspective: across a wide range of examples, a single non-random quantum state
is shown to encode universal and highly random quantum state ensembles. We
characterize these ensembles using the notion of quantum state $k$-designs from
quantum information theory and investigate their universality using a
combination of analytic and numerical techniques. In particular, we establish
that $k$-designs arise naturally from generic states as well as individual
states associated with strongly interacting, time-independent Hamiltonian
dynamics. Our results offer a new approach for studying quantum chaos and
provide a practical method for sampling approximately uniformly random states;
the latter has wide-ranging applications in quantum information science from
tomography to benchmarking.
