Loschmidt echo and scrambling of systematic errors in tomography-A quantum signature of chaos

How does quantum chaos lead to rapid scrambling of information as well as systematic 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. We construct a quantity to capture the scrambling of systematic errors, an out-of-time-ordered correlator (OTOC), which serves as a signature of chaos and quantifies the spread of errors. We show that the spread of errors, as quantified by the OTOC, is related to the operator Loschmidt echo, which is defined as the Hilbert-Schmidt inner product of the operators On, and O'n generated from repeated application of the Floquet map for ideal (unperturbed) dynamics and the true (perturbed) dynamics, respectively. This also gives us an operational interpretation of Loschmidt echo (LE) for operators by connecting it to the performance of quantum tomography. We show how our results demonstrate not only a link between LE and scrambling of errors different than previous studies, but also that such a link can have operational consequences in quantum information processing.