Quantum chaos is the study of signatures of classical chaos in corresponding quantum system(s). We adopt a continuous weak measurement tomography protocol to explore the signatures of chaos. We generate the measurement record as a series of expectation values of an observable evolving under the desired dynamics, which can show a transition from integrability to complete chaos. We find that the rate of information gain, and hence, the fidelity obtained in tomography, depends not only on the degree of chaos in the dynamics and to what extent it causes the initial observable to spread in various directions of the operator space but, more importantly, on how well these directions are aligned with the density matrix to be estimated.

We further investigate operator spreading and chaos in the Krylov subspace. We study operator spreading in many-body quantum systems by its potential to generate an informationally complete measurement record in quantum tomography for random states. We find that the amount of operator spreading, as quantified by the fidelity in quantum tomography and various other metrics of information gain, increases with the degree of chaos in the system. We consider two many-body systems: the tilted field Ising model with and without delta kicks and the Heisenberg XXZ model with a single impurity to explore operator spreading. We find our approach in quantifying operator spreading is a more consistent indicator of quantum chaos than Krylov complexity as the latter may correlate/anti-correlate or show no explicit behavior with the level of chaos in the dynamics. Our study also gives an operational interpretation for operator spreading in terms of fidelity gain in an actual quantum information tomography protocol.

Continuing in our journey of finding the footprints of chaos in the quantum domain, we explore the growth of errors in noisy tomography. Interestingly, we find that the reconstruction fidelity initially increases regardless of the degree of chaos or the strength of perturbations in the dynamics. For random states, when the measurement record is obtained from a random initial observable, the subsequent drop in the fidelity obtained is inversely correlated to the degree of chaos in the dynamics. More importantly, this also gives us an operational interpretation of Loschmidt echo for operators by connecting it to the performance of quantum tomography. We find a quantity to capture the scrambling of errors, an out-of-time-ordered correlator (OTOC) between two operators under perturbed and unperturbed system dynamics that serves as a signature of chaos and quantifies the spread of errors. Our results demonstrate not only a fundamental link between Loschmidt echo and scrambling of errors, as captured by OTOCs, but that such a link can have operational consequences in quantum information processing.