  How does quantum chaos lead to rapid scrambling of information as well as
errors across a system when one introduces perturbations in the dynamics? What
are its consequences for the reliability of quantum simulations and quantum
information processing? We employ continuous measurement quantum tomography as
a paradigm to study these questions. The measurement record is generated as a
sequence of expectation values of a Hermitian observable evolving under
repeated application of the Floquet map of the quantum kicked top.
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 define
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.
