  We show that the most important measures of quantum chaos like frame
potentials, scrambling, Loschmidt echo, and out-of-time-order correlators
(OTOCs) can be described by the unified framework of the isospectral twirling,
namely the Haar average of a $k$-fold unitary channel. We show that such
measures can then be always cast in the form of an expectation value of the
isospectral twirling. In literature, quantum chaos is investigated sometimes
through the spectrum and some other times through the eigenvectors of the
Hamiltonian generating the dynamics. We show that, by exploiting random matrix
theory, these measures of quantum chaos clearly distinguish the finite time
profiles of probes to quantum chaos corresponding to chaotic spectra given by
the Gaussian Unitary Ensemble (GUE) from the integrable spectra given by
Poisson distribution and the Gaussian Diagonal Ensemble (GDE). On the other
hand, we show that the asymptotic values do depend on the eigenvectors of the
Hamiltonian. We see that the isospectral twirling of Hamiltonians with
eigenvectors stabilizer states does not possess chaotic features, unlike those
Hamiltonians whose eigenvectors are taken from the Haar measure. As an example,
OTOCs obtained with Clifford resources decay to higher values compared with
universal resources. Finally, we show a crossover in the OTOC behavior between
a class of integrable models and quantum chaos.

Quantum Physics