Uniform observable error bounds of Trotter formulae for the semiclassical Schrödinger equation

TL;DR

This work addresses the multiscale challenge of simulating the semiclassical Schrödinger equation by proving uniform-in-$h$ observable error bounds for both first- and second-order Trotter formulae, thereby achieving $O(1)$ Trotter steps for certain observables without sacrificing convergence order. The authors develop and combine semiclassical Weyl calculus with discrete microlocal analysis to extend the results from the continuous setting to spatial discretizations, providing a rigorous bridge to finite-dimensional quantum simulations. Key contributions include direct, microscopic error analysis for observable evolution (not just state norms), and the first application of discrete microlocal analysis to quantum computation, yielding $h$-independent query complexities for observable calculations. The findings illuminate how multiscale structure, via the small parameter $h$, can be exploited to improve quantum dynamics simulation and inform design of quantum algorithms for semiclassical regimes with high precision requirements.

Abstract

Known as no fast-forwarding theorem in quantum computing, the simulation time for the Hamiltonian evolution needs to be $O(\|H\| t)$ in the worst case, which essentially states that one can not go across the multiple scales as the simulation time for the Hamiltonian evolution needs to be strictly greater than the physical time. We demonstrated in the context of the semiclassical Schrödinger equation that the computational cost for a class of observables can be much lower than the state-of-the-art bounds. In the semiclassical regime (the effective Planck constant $h \ll 1$), the operator norm of the Hamiltonian is $O(h^{-1})$. We show that the number of Trotter steps used for the observable evolution can be $O(1)$, that is, to simulate some observables of the Schrödinger equation on a quantum scale only takes the simulation time comparable to the classical scale. In terms of error analysis, we improve the additive observable error bounds [Lasser-Lubich 2020] to uniform-in-$h$ observable error bounds. This is, to our knowledge, the first uniform observable error bound for semiclassical Schrödinger equation without sacrificing the convergence order of the numerical method. Based on semiclassical calculus and discrete microlocal analysis, our result showcases the potential improvements taking advantage of multiscale properties, such as the smallness of the effective Planck constant, of the underlying dynamics and sheds light on going across the scale for quantum dynamics simulation.

Figures (3)

• Figure 1: Log-log plot of the scaling of the operator norm of the difference of the observable evolution matrices $T_\ell(s) - T(s)$ (for $\ell = 1,2$) for various time step size $s$. "Trotter1" labels the first-order Trotter formula while "Trotter2" labels the second-order Trotter formula. "Ob1" is for the cosine observable that is the quantization of $\cos(x)$ and "Ob2" denotes the momentum observable $\hat{p} = -\mathrm{i} h \nabla_x$. The reference line is for asymptotic scaling in $s$.
• Figure 2: Log-log plot of the errors in the operator norm for various Planck constants $h$. "unitary" denotes the error measuring in the operator norm of the unitaries, while "ob1" and "ob2" measures the operator norm error of the observable evolution matrices. The errors in the unitary scales as $1/h$. However, observable errors do not grow as $h$ decreases.
• Figure 3: Long-time error in the operator norm of the unitaries and the observable for various $s$ (Left) or $h$ (Right). The final time is $1$. "unitary" denotes the error measuring in the operator norm of the unitaries, while "ob1" and "ob2" measure the operator norm error of the observable evolution matrices. For both Trotter formulae, the long-time errors are uniform in $h$. The first-order Trotter formula has a long-time observable error bound of $\mathcal{O}(s)$ and the second-order one exhibits $\mathcal{O}(s^2)$.

Theorems & Definitions (18)

• Lemma 1: Variation of parameter formula
• Theorem 2: Local error of observable for Trotter1
• proof
• Theorem 3: Local error of observable for Trotter2
• proof
• Corollary 4
• Theorem 5
• Theorem 6
• proof : Proof of Theorem \ref{['thm:long_trot2']}
• Remark 7