Nonadiabatic Self-Healing of Trotter Errors in Digitized Counterdiabatic Dynamics

TL;DR

This work shows that finite-time self-healing of Trotter errors persists when diabatic transitions are suppressed via counterdiabatic driving, even away from the adiabatic limit. By mapping the residual digitization error to a weak harmonic perturbation in the instantaneous eigenbasis, the authors derive an analytic upper bound on the finite-time Trotter error and reveal a phase-cancellation mechanism that drives oscillatory, bounded error accumulation. The analysis is validated across noninteracting and interacting spin models, including Ising chains and fully connected p-spin systems, using exact and variational CD constructions. The results offer practical guidance for high-fidelity state preparation on gate-based quantum processors, highlighting a co-design path: first minimize diabatic errors with CD, then tune the Trotter step and schedule to exploit interference and digitization limitations. Overall, finite-time self-healing emerges as an intrinsic feature of digitized counterdiabatic protocols with broad relevance to digital quantum simulation and state preparation on NISQ devices.

Abstract

Trotter errors in digitized quantum dynamics arise from approximating time-ordered evolution under noncommuting Hamiltonian terms with a product formula. In the adiabatic regime, such errors are known to exhibit long-time self-healing [Phys. Rev. Lett. \textbf{131}, 060602 (2023)], where discretization effects are effectively suppressed. Here we show that self-healing persists at finite evolution times once nonadiabatic errors induced by finite-speed ramps are compensated. Using counterdiabatic driving to cancel diabatic transitions and isolate discretization effects, we study both noninteracting and interacting spin models and characterize the finite-time scaling with the Trotter steps and the total evolution time. In the instantaneous eigenbasis of the driven Hamiltonian, the leading digital error maps to an effective harmonic perturbation whose dominant Fourier component yields an analytic upper bound on the finite-time Trotter error and reveals the phase-cancellation mechanism underlying self-healing. Our results establish finite-time self-healing as a generic feature of digitized counterdiabatic protocols, clarify its mechanism beyond the long-time adiabatic limit, and provide practical guidance for high-fidelity state preparation on gate-based quantum processors.

Figures (5)

• Figure 1: (a, b) Infidelity versus evolution time $T$ for $\Delta t=0.1$ (purple line), $\Delta t= 0.01$ (yellow line) and $\Delta t= 0.001$ (blue line), for a single qubit. In panel (a), there are no counterdiabatic corrections. In panel (b), we include the exact CD potential. The axes are in logarithmic scale. (c, d) Dynamics of the infidelity, $\Delta t=0.01$. In these plots, we show the dynamics for a single qubit. Blue curves represent the dynamics without CD corrections, orange curves represent the dynamics with the exact CD potential. In panel (c), $T=1$; in panel (d), $T=100$.
• Figure 2: Infidelity versus final time $T$ for the Ising model with the exact CD potential. Orange triangles represent data for $\Delta t = 0.01$, while purple circles correspond to $\Delta t=0.1$. Continuous lines represent numerical fits done using the analytical prediction given by Eq. \ref{['eq:perturbation-theory-T']}. (a) $J_Z = 0.1$ (b) $J_Z = 0.5$ (c) $J_Z=1$. (d) Infidelity vs $T$ for $\Delta t=0.001$ for $J_Z = 0.1$. In these plots we consider a system size $N=6$.
• Figure 3: Final infidelity $\mathcal{I}(T)$ for the $p$-spin ($p=2$) model, $\Delta t=0.01$. (a) $N=10$ (b) $N=30$. Different points correspond to different approximation orders $l$ of the CD potential in the nested commutator expansion.
• Figure 4: Infidelity $\mathcal{I}(t)$ versus time for $t\in\left[0,T\right]$, for $T=1$. The evolution is for a $p$-spin model for $p=2$ and size $N=10$. The evolution is given by Eq. \ref{['eq:CD-digitalized-evolution']} with the variational CD potential, for different orders $l$ of approximation in the nested commutator expansion. (a) $\Delta t=0.1$. (b) $\Delta t= 0.01$. (c) $\Delta t= 0.001$. We see that, as the order of approximation increases, for $t\rightarrow T$, $\mathcal{I}(T)$ approaches zero for all $\Delta t$.
• Figure 5: Infidelity $\mathcal{I}(T)$ as a function of the time step $\Delta t$. Violet points correspond to the Ising model with $N=6$ and $J_Z=0.1$. Yellow points represent the noninteracting case. Green points refer to the interacting $p$-spin model with $p=2$. The blue line indicates the expected $\Delta t^2$ scaling of the Trotter error. Both the evolution and the Trotterization are performed with the inclusion of the CD potential. The total evolution time is fixed at $T=1$. The axes are in logarithmic scale.