Trotterization is substantially efficient for low-energy states

TL;DR

This work establishes an optimal, initial-state–dependent bound for Trotterization in simulating local quantum Hamiltonians. By projecting onto the low-energy subspace and bounding nested commutators with a Δ-dependent, locality-aware analysis, the authors show the Trotter error scales linearly with the initial energy Δ (up to small leaks) and only polylogarithmically with system size N, yielding substantial cost reductions for low-energy states across both finite-range and certain long-range interactions. They derive exact Trotter-number scaling in terms of Δ, t, N, ε, and the Trotter order p, and validate the theory numerically on frustration-free models, demonstrating dramatic improvements over generic initial states. The results further extend to weakly-correlated states with low-energy expectations, via concentration bounds, indicating broad applicability to realistic physical states in condensed matter and quantum chemistry. Overall, the paper provides a theoretically optimal framework for state-aware Hamiltonian simulation, enabling fast and accurate dynamics for low-energy physics.

Abstract

Trotterization is one of the central approaches for simulating quantum many-body dynamics on quantum computers or tensor networks. In addition to its simple implementation, recent studies have revealed that its error and cost can be reduced if the initial state is closed in the low-energy subspace. However, the improvement by the low-energy property rapidly vanishes as the Trotter order grows in the previous studies, and thus, it is mysterious whether there exists genuine advantage of low-energy initial states. In this Letter, we resolve this problem by proving the optimal error bound and cost of Trotterization for low-energy initial states. For generic local Hamiltonians composed of positive-semidefinite terms, we show that the Trotter error is at most linear in the initial state energy $Δ$ and polylogarithmic in the system size $N$. As a result, the computational cost becomes substantially small for low-energy states with $Δ\in o(Ng)$ compared to the one for arbitrary initial states, where $g$ denotes the energy per site and $Ng$ means the whole-system energy. Our error bound and cost of Trotterization achieve the theoretically-best scaling in the initial state energy $Δ$. In addition, they can be partially extended to weakly-correlated initial states having low-energy expectation values, which are not necessarily closed in the low-energy subspace. Our results will pave the way for fast and accurate simulation of low-energy states, which are one central targets in condensed matter physics and quantum chemistry.

Figures (2)

• Figure S1: System size dependence of the Trotter error in the AKLT and MG Hamiltonians. The blue lines denote the error for arbitrary initial states, $\norm{e^{-iHt}-T_p(t)}$. The green or orange lines show the errors for initial states in the low-energy subspace, $\norm{(e^{-iHt}-T_p(t)) \Pi_{\leq \Delta}}$. The integer $p$ denotes the Trotter order.
• Figure S2: $\Delta$-dependence of the Trotter error, where $\Delta$ denotes the maximal energy scale of the initial state.

Theorems & Definitions (6)

• Theorem 1
• Theorem S1
• Corollary S2
• Theorem S3
• Corollary S4
• Proposition S5