How classical chaos emerges from the underlying quantum world is a fundamental problem in physics. The origin of this question is in the correspondence principle, which states that quantum mechanics, the physical theory of microscopic objects, should reproduce classical laws in the macroscopic limit. By studying a quantum system whose classical analogue is chaotic, we can understand the quantum features that lead to chaos in the classical limit. We use kicked tops of a few qubits in the deep quantum regime to investigate the onset of chaos. We analyze a couple of the dynamical diagnostics of chaos called OTOCs and Loschmidt echo in this study. We find residual signatures of classical chaos even in the deep quantum regime. We also provide an exponentially efficient protocol to measure OTOCs, a quantifier of chaos, using a single pure qubit quantum computation (DQC1) protocol. This protocol also helps to benchmark unitary gates, which is important from the perspective of quantum computation and quantum control.

Another domain where one can study the effects of chaos is quantum state tomography. Quantum state reconstruction is a nontrivial problem because of the inherent quantum uncertainty. Traditional projective measurements to find out information about the state require an infinite number of copies of the system. We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators generated by a unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step. We quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of the covariance matrix of the estimation. Surprisingly, high fidelity of reconstruction is obtained using random unitaries diagonal in a fixed basis even though the measurement record is not informationally complete. We also give an upper bound on the maximal Fisher information that can be obtained in tomography using the Wishart-Laguerre ensemble of random matrices and the associated Marchenko-Pastur distribution.The effect of chaotic dynamics in tomography is also investigated. The typicality of random states leads to a brief discussion on the concentration of measure phenomena.