We establish tight connections between entanglement entropy and the approximation error in Trotter-Suzuki product formulas for Hamiltonian simulation. Product formulas remain the workhorse of quantum simulation on near-term devices, yet standard error analyses yield worst-case bounds that can vastly overestimate the resources required for structured problems. For systems governed by geometrically local Hamiltonians with maximum entanglement entropy $S_\text{max}$ across all bipartitions, we prove that the first-order Trotter error scales as $\mathcal{O}(t^2 S_\text{max} \operatorname{polylog}(n)/r)$ rather than the worst-case $\mathcal{O}(t^2 n/r)$, where $n$ is the system size and $r$ is the number of Trotter steps. This yields improvements of $\tildeΩ(n^2)$ for one-dimensional area-law systems and $\tildeΩ(n^{3/2})$ for two-dimensional systems. We extend these bounds to higher-order Suzuki formulas, where the improvement factor involves $2^{pS^*/2}$ for the $p$-th order formula. We further establish a separation result demonstrating that volume-law entangled systems fundamentally require $\tildeΩ(n)$ more Trotter steps than area-law systems to achieve the same precision. This separation is tight up to logarithmic factors. Our analysis combines Lieb-Robinson bounds for locality, tensor network representations for entanglement structure, and novel commutator-entropy inequalities that bound the expectation value of nested commutators by the Schmidt rank of the state. These results have immediate applications to quantum chemistry, condensed matter simulation, and resource estimation for fault-tolerant quantum computing.

翻译：我们建立了纠缠熵与哈密顿量模拟中Trotter-Suzuki乘积公式近似误差之间的紧密联系。乘积公式仍是近期量子模拟设备的核心方法，然而标准误差分析给出的最坏情况界可能严重高估结构化问题所需的资源。对于所有二分划分下最大纠缠熵为$S_\text{max}$的几何局域哈密顿量系统，我们证明一阶Trotter误差按$\mathcal{O}(t^2 S_\text{max} \operatorname{polylog}(n)/r)$标度变化，而非最坏情况的$\mathcal{O}(t^2 n/r)$，其中$n$为系统尺寸，$r$为Trotter步数。这为一维面积律系统带来$\tildeΩ(n^2)$的改进，为二维系统带来$\tildeΩ(n^{3/2})$的改进。我们将这些界推广至高阶Suzuki公式，其中$p$阶公式的改进因子涉及$2^{pS^*/2}$。我们进一步建立了分离结果，证明体积律纠缠系统要达到相同精度，本质上比面积律系统需要多$\tildeΩ(n)$个Trotter步数，该分离结果在对数因子内是紧致的。我们的分析结合了局域性的Lieb-Robinson界、纠缠结构的张量网络表示，以及通过态Schmidt秩界定嵌套对易子期望值的新型对易子-熵不等式。这些结果对量子化学、凝聚态模拟和容错量子计算的资源估计具有直接应用价值。