Uniform semiclassical observable error bound of Trotter-Suzuki splitting: a simple algebraic proof

TL;DR

The paper tackles the challenge of achieving uniform-in-$h$ observable error bounds for high-order Trotter-Suzuki splittings applied to the semiclassical Schrödinger equation, where wavefunction errors typically depend on the semiclassical parameter $h$. It introduces a simple algebraic, Egorov-free framework based on Taylor expansions and nested commutator estimates, applicable to both continuous operators and spatial discretizations. The main result is a global bound of the form $\| T_{p,n}(\Delta t) - T(t) \| \le C\,\Delta t^{p}$ with a constant $C$ independent of $h^{-1}$, together with a suite of nested-commutator bounds that remain $h$-independent under spectral and finite-difference discretizations. Numerical experiments validate the theoretical rates and reinforce the $h$-independence of the observable error, highlighting the practical impact for efficient, high-order quantum simulations of semiclassical observables. Overall, the work provides a broadly applicable, algebraic toolkit for analyzing Trotterized semiclassical dynamics without resorting to semiclassical limit techniques, with implications for quantum simulation and numerical analysis.

Abstract

Efficient simulation of the semiclassical Schrödinger equation has garnered significant attention in the numerical analysis community. While controlling the error in the unitary evolution or the wavefunction typically requires the time step size to shrink as the semiclassical parameter $h$ decreases, it has been observed -- and proved for first- and second-order Trotterization schemes -- that the error in certain classes of observables admits a time step size independent of $h$. In this work, we explicitly characterize this class of observables and present a new, simple algebraic proof of uniform-in-$h$ error bounds for arbitrarily high-order Trotterization schemes. Our proof relies solely on the algebraic structure of the underlying operators in both the continuous and discrete settings. Unlike previous analyses, it avoids Egorov-type theorems and bypasses heavy semiclassical machinery. To our knowledge, this is the first proof of uniform-in-$h$ observable error bounds for Trotterization in the semiclassical regime that relies only on algebraic structure, without invoking the semiclassical limit.

Figures (4)

• Figure 1: This figure illustrates the distinct proof strategy employed in our work compared to prior significant advancements BaoJinMarkowich2002JinMarkowichSparber2011GolseJin19LasserLubich2020FangTres2021JinLiLiu2021 on this topic. Rather than using macroscopic (semiclassical) limits as an intermediate step and controlling the total error via a decomposition into multiple contributions, we directly estimate the error in the time-evolved observable produced by the numerical scheme. $E_1$, $E_2$, and $E_3$ denote the errors between the two objects they connect, measured in an appropriate error metric (see the cited work above for further details). Different from the previous analysis developed by one of the authors for first- and second-order Trotterizations, this work develops a new purely algebraic proof and extends the uniform-in-$h$ error bounds to arbitrarily high-order Trotterization schemes.
• Figure 1: Log-log plots showing the convergence of errors with respect to the step size $\Delta t$ for 1st, 2nd, 4th, and 6th-order Trotter methods. (a) Observable errors are plotted against $\Delta t$, demonstrating convergence consistent with the expected asymptotic scaling. (b) Unitary errors are plotted against $\Delta t$, also showing the convergence rate aligning with the order of the Trotter method. The $\Delta t^p$ reference lines provide a visual confirmation of the expected scaling behavior for each method.
• Figure 2: Log-log plots showing unitary and observable errors as a function of the semiclassical parameter $h$ for 2nd, 4th, and 6th-order Trotter methods. Each plot demonstrates that while the unitary error scales as $h^{-1}$ (indicated by the reference $h^{-1}$ line), the observable error remains independent of $h$, with no apparent slope. This highlights the observable error’s insensitivity to the semiclassical parameter $h$, particularly scaling independently with $h^{-1}$ aligning with our theoretical results.
• Figure 3: Log-log plot of commutators illustrating the scaling behavior of nested commutators with observables. The norm of the first commutator $[A, B]$ scales as ${\mathcal{O}}(1/h)$, confirming the dependence on the semiclassical parameter $h^{-1}$. In contrast, higher-order nested commutators with the observable $O$, such as $[[A, B], O]$, $[A, [[A, B], O]]$, and $[A, [A, [[A, B], O]]]$, demonstrate independence from $h^{-1}$ and remain consistent with ${\mathcal{O}}(1)$ scaling. The observed behavior aligns with theoretical predictions of observable error insensitivity to $h^{-1}$ in higher-order Trotter approximations.

Theorems & Definitions (26)

• Theorem 2.1: Global observable error for $p$-th order Trotterization
• Remark 2.2: On the Dimensionality
• Corollary 2.3: Query Complexity of Trotter-Suzuki Splittings
• Remark 2.4
• Proposition 2.5: General observable error bound for $p$-th order Trotter-Suzuki formula
• Theorem 3.1: Local error of observable for $p$-th order Trotter
• Lemma 4.1: Height-Reduction and Width Expansion
• Lemma 4.2
• Proof 1
• Theorem 4.3