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In this work, we provide an analytical proof of the robustness of a form of topological entanglement under a model of random local perturbations. We define the notion of topological purity and show that, in the context of quantum double models, this quantity does detect topological order and is robust under the action of a random shallow quantum circuit. 

We show that the most important measures of quantum chaos, such as 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 always be 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 thanks to this technique, we can interpolate smoothly between integrable Hamiltonians and quantum chaotic Hamiltonians. The isospectral twirling of Hamiltonians with eigenvector 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. By doping Hamiltonians with non-Clifford resources, we show a crossover in the OTOC behavior between a class of integrable models and quantum chaos. Moreover, exploiting random matrix theory, we show that these measures of quantum chaos clearly distinguish the finite time behavior 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). 

It is well known that a quantum circuit on N qubits composed of Clifford gates with the addition of k non Clifford gates can be simulated on a classical computer by an algorithm scaling as poly ( N ) exp ⁡ ( k ) \cite{bravyi2016improved}. We show that, for a quantum circuit to simulate quantum chaotic behavior, it is both necessary and sufficient that k = Θ ( N ) . This result implies the impossibility of simulating quantum chaos on a classical computer. 

We present a systematic construction of probes into the dynamics of isospectral ensembles of Hamiltonians by the notion of Isospectral twirling, expanding the scopes and methods of ref. [1]. The relevant ensembles of Hamiltonians are those defined by salient spectral probability distributions. The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal Ensemble (GDE) describe non chaotic, integrable dynamics. We compute the Isospectral twirling of several classes of important quantities in the analysis of quantum many-body systems: Frame potentials, Loschmidt Echos, OTOCs, Entanglement, Tripartite mutual information, coherence, distance to equilibrium states, work in quantum batteries and extension to CP-maps. Moreover, we perform averages in these ensembles by random matrix theory and show how these quantities clearly separate chaotic quantum dynamics from non chaotic ones.