Scrambling of Entanglement from Integrability to Chaos: Bootstrapped Time-Integrated Spread Complexity

A new general analytical relationship between spread complexity and fidelity of quantum dynamics is established with time-integrated quantities under operator perturbation. This approach diagnoses the degree of quantum ergodicity and the behavior of scrambling of entanglement simultaneously. The equivalence between the perturbation scheme and bootstrapped Hamiltonian realizations is shown to be a rigorous testbed. Using bootstrapped Rosenzweig-Porter ensembles for unitary dynamical perturbation, we demonstrated that the integrated spread complexity provides a fine-grained resolution across different ergodic regimes, exhibiting a monotonic inverse relationship with the decay of integrated fidelity for maximally entangled states.