Trotter Scars: Trotter Error Suppression in Quantum Simulation

Abstract

Recent studies have shown that Trotter errors are highly initial-state dependent and that standard upper bounds often substantially overestimate them. However, the mechanism underlying anomalously small Trotter errors and a systematic route to identifying error-resilient states remain unclear. Using interaction-picture perturbation theory, we derive an analytical expression for the leading-order Trotter error in the eigenbasis of the Hamiltonian. Our analysis shows that initial states supported on spectrally commensurate energy ladders exhibit strongly suppressed error growth together with persistent Loschmidt revivals. We refer to such states as Trotter scars. To identify such states in practice, we further introduce a general variational framework for finding error-minimizing initial states for a given Hamiltonian. Applying this framework to several spin models, we find optimized states whose spectral support and dynamical behavior agree with the perturbative prediction. Our results reveal the spectral origin of Trotter-error resilience and provide a practical strategy for discovering error-resilient states in digital quantum simulation.

Figures (2)

• Figure 1: Characterization of Trotter scars in the Heisenberg chain (first column, $h_x=0.5$), the Stark spin chain (second column, $h_x=0.8$, $h_y=0.9$, $h_z=4.0$), and the PXP model (third column). (a,b,c) Loschmidt echo $\mathcal{F}(t)$ for the optimized state and 100 random states. The vertical dashed line marks the stroboscopic time $t_1=2\pi/\Omega$. (d,e,f) Accumulated Trotter error $\|\delta\psi(t)\|$ for the optimized state and the random-state average over 100 initial states. The black dotted lines are the perturbative result obtained from Eq. \ref{['eq:error_norm_spectral']}, which agrees well with the exact Trotter error. (g,h,i) Eigenstate overlap $|\langle n|\psi_{\mathrm{opt}}\rangle|^2$ versus eigenenergy $E_n$ for the 20 eigenstates with the largest overlap, showing that the optimized state concentrates its spectral weight on equally spaced energy levels, consistent with the commensurability condition predicted by Eq. \ref{['eq:error_norm_spectral']}. In (g), eigenstates are labeled by their total spin quantum number $S$ obtained from simultaneous diagonalization of $H$, $\mathbf{S}^2$, and $S^x$. All results are for $L=12$ with optimization time $T_l=10$ and Trotter step $\Delta t=0.01$.
• Figure 2: Bloch-sphere trajectories of $(\langle\sigma^x_{11}\rangle,\langle\sigma^y_{11}\rangle,\langle\sigma^z_{11}\rangle)$ under exact time evolution for (a) the Heisenberg chain and (b) the PXP model, comparing the optimized Trotter-scar state (green), a Haar-random product state (blue), and the Néel state $|\mathbb{Z}_2\rangle$ (orange, PXP only). In both models, the optimized state traces a simple closed orbit, directly reflecting the periodic revivals of the global wave function. By contrast, random states follow irregular trajectories. In the PXP model, the $|\mathbb{Z}_2\rangle$ state follows nested spiraling orbits whose amplitude gradually decays, mirroring the slow degradation of its Loschmidt-echo revivals. Parameters are the same as in Fig. \ref{['fig:trotter_error']}.