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

Abstract

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.

Figures (4)

• Figure 1: Schematic of the multi-round time-reversed dynamics, illustrated using the example of two rounds. Taking decoherence effects into account, the forward evolution is described by a Lindbladian superoperator $\mathcal{L}_H$ with Hamiltonian $H$. The Hamiltonian in the backward evolution, denoted by $\tilde{H}$, exhibits small deviations from $H$. These coherent and incoherent errors prevent perfect time reversal, with their effects being amplified by the quantum butterfly effect.
• Figure 2: Graphical representation of the Loschmidt echo $F_n(t)$ for (a) $n=1$ and (b) $n=2$. Branches with the same color originate from the same Lindbladian evolution. Solid lines correspond to evolution governed by $\mathcal{L}_H$, while dotted lines indicate the Lindblad term. Double lines denote evolution in the presence of coherent errors, where insertions of $\delta H$, marked by red dots, may occur. In panel (b), all branches are labeled from 1 to 8.
• Figure 3: Typical scramblon diagrams contributing to the Loschmidt echo with $n=2$ for (a) incoherent errors and (b) coherent errors. Wavy lines represent scramblon propagators, while solid lines denote microscopic operators. Subscripts on the error operators indicate their branch indices (see FIG. \ref{['fig:contour']}). In panel (a), all scramblons emitted by jump operators are absorbed by the operator $O$. In panel (b), scramblons emitted by inter-round errors with $t_1,t_2,t_3,t_4\approx t$ can also be absorbed by errors with $t_5, t_6\approx0$.
• Figure 4: Numerical demonstration of our predictions using the solvable SYK model for the $n$-round Loschmidt echo with either coherent or incoherent errors at $V/J=0.01$. The data points represent results from numerical simulations, while the solid lines are fits based on Eq. \ref{['eqn:resIn']}, and the dashed lines are plotted using Eq. \ref{['eqn:resCo']} with fitted parameters $\gamma_{c}=\gamma_{I}=5.85\times10^{-4}, \varkappa=0.866, \Delta_{O}=1.37$. The results clearly demonstrate a crossover from quadratic to linear scaling for coherent errors.