Distinguishing Coherent and Incoherent Errors in Multi-Round Time-Reversed Dynamics via Scramblons

Despite the rapid development of quantum science and technology, errors are inevitable and play a crucial role in quantum simulation and quantum computation. In quantum chaotic systems, coherent errors arising from imperfect Hamiltonian control and incoherent errors induced by coupling to the environment are both exponentially amplified during time evolution due to information scrambling. A fundamental question is how these two classes of errors imprint distinct signatures on the emergent irreversibility of many-body dynamics. In this Letter, we address this question by investigating multi-round time-reversed dynamics in the presence of both coherent and incoherent errors. By applying scramblon theory, we obtain closed-form expressions for the Loschmidt echo over different rounds of time-reversed evolution. For incoherent errors, the error accumulates linearly with the number of rounds, whereas coherent errors exhibit a crossover from quadratic to linear accumulation. These predictions are explicitly verified using the solvable Sachdev-Ye-Kitaev model. Our results provide a theoretical foundation for characterizing and calibrating coherent and incoherent errors in reversed dynamics, with particular relevance to nuclear magnetic resonance systems.