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arXiv:2512.22636 · Physical Review A

Nonadiabatic Self-Healing of Trotter Errors in Digitized Counterdiabatic Dynamics

Mara Vizzuso, Gianluca Passarelli, Giovanni Cantele, Procolo Lucignano, Xi Chen, Koushik Paul

quant-phSubmitted Dec 27, 2025• Announced Dec 30, 2025• 0 citations
v1 · Dec 27, 2025v2 · Jun 12, 2026
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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.

Journal reference
Phys. Rev. A 113, 062432 (2026)
DOI
10.1103/5ck5-tp69

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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.
Adiabatic time evolution can be used to prepare a complicated quantum many-body state from one that is easier to synthesize and Trotterization can be used to implement such an evolution digitally. The complex interplay between nonadiabaticity and digitization influences the infidelity of this process. We prove that the first-order Trotterization of a complete adiabatic evolution has a cumulative infidelity that scales as O(T^{-2}δt^{2}) instead of O(T^{2}δt^{2}) expected from general Trotter error bounds, where δt is the time step and T is the total time. This result suggests a self-healing mechanism and explains why, despite increasing T, infidelities for fixed-δt digitized evolutions still decrease for a wide variety of Hamiltonians. It also establishes a correspondence between the quantum approximate optimization algorithm and digitized quantum annealing.
Adiabatic quantum computing is a general framework for preparing eigenstates of Hamiltonians on quantum devices. However, its digital implementation requires an efficient Hamiltonian simulation subroutine, which may introduce extra computational overhead or complicated quantum control logic. In this work, we show that the time-step sizes in time discretization can be much larger than expected, and the overall complexity is greatly reduced. Remarkably, regardless of the general convergence order of the numerical method, we can choose a uniform time-step size independent of tolerated error and evolution time for sufficiently accurate simulation. Furthermore, with the boundary cancellation condition where the continuous diabatic errors are exponentially suppressed, we provide strong evidence for a superpolynomial convergence of first-order Trotter with uniform time-step size. We apply our analysis to the example of adiabatic unstructured search and show several preferable features of the Trotterized adiabatic approach: It can match the Grover lower bound, it does not require knowledge of the number of marked states, and its performance can be asymptotically comparable with that of the quantum approximate optimization algorithm.
The simulation of adiabatic evolution has deep connections with adiabatic quantum computation, the quantum approximate optimization algorithm, and adiabatic state preparation. Here we address the error analysis problem in quantum simulation of adiabatic process using Trotter formulas. We show that with additional conditions, the circuit depth can be linear in simulation time TTT. The improvement comes from the observation that the fidelity error here can't be estimated by the norm distance between evolution operators. This phenomenon is termed the robustness of discretization in digital adiabatic simulation. It can be explained in three steps, from analytical and numerical evidence: (1) The fidelity error should be estimated by applying adiabatic theorem on the effective Hamiltonian instead. (2) Because of the specialty of Riemann-Lebesgue lemma, most adiabatic process is naturally robust against discretization. (3) As the Trotter step gets larger, the spectral gap of effective Hamiltonian tends to close, which results in the failure of digital adiabatic simulation.
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, we further introduce a model-agnostic variational framework. Its loss function can be built from Trotterized dynamics alone, which allows the search to reach system sizes beyond exact diagonalization. The optimized states at small sizes moreover follow regular patterns that extend to larger sizes. We demonstrate our theory in three spin models, where the optimized states exhibit the predicted persistent Loschmidt revivals and strongly suppressed error growth. We further conducted experiments on a 171717-qubit superconducting quantum processor and successfully realized the Trotter-scar states and demonstrated the Trotter error suppression in quantum simulations.
Digitizing an adiabatic evolution is a strategy able to combine the good performance of gate-based quantum processors with the advantages of adiabatic algorithms, thus providing a hybrid model for efficient quantum information processing. In this work we develop validity conditions for high-fidelity digital adiabatic tasks. To this end, we assume a digitizing process based on the Suzuki-Trotter decomposition, which allows us to introduce a digitized adiabatic theorem. As a consequence of this theorem, we show that the performance of such a hybrid model is limited by the fundamental constraints on the adiabatic theorem validity, even in ideal quantum processors. We argue how our approach predicts the existence of intrinsic nonadiabatic errors reported by Barends et al. [Nature (London) 534, 222 (2016)] through an empirical study of digital annealing. In addition, our approach allows us to explain the existence of a scaling of the number of Suzuki-Trotter blocks for the optimal digital circuit with respect to the optimal adiabatic total evolution time, as reported by Mbeng et al. [Phys. Rev. B 100, 224201 (2019)] through robust numerical analysis of digital annealing. We illustrate our results through two examples of digitized adiabatic algorithms, namely, the two-qubit exact-cover problem and the three-qubit adiabatic factorization of the number 21.
A system undergoes adiabatic evolution when its population in the instantaneous eigenbasis of its time-dependent Hamiltonian changes only negligibly. The realization of such dynamics requires slow-enough changes of the parameters of the Hamiltonian, a task that can be hard to achieve near quantum critical points. A powerful alternative is provided by the counterdiabatic modification of the Hamiltonian allowing for an arbitrarily quick implementation of the adiabatic dynamics. Such a counterdiabatic driving protocol has been recently proposed for the quantum Ising model (del Campo et al 2012 Phys. Rev. Lett. 109 115703 ). We derive an exact closed-form expression for all the coefficients of the counterdiabatic Ising Hamiltonian. We discuss two approximations to the exact counterdiabatic Ising Hamiltonian quantifying their efficiency of the dynamical preparation of the desired ground state. In particular, these studies show how quantum criticality enhances finite-size effects in counterdiabatic dynamics.