AI 中文总结
本文针对分离能标的哈密顿量，提出两类随机化乘积公式，在两种访问模型下分别实现$\mathcal O(\alpha^2)$和$\mathcal O(\alpha^3)$误差标度，优于确定性公式下界，数值模拟验证了门数量的减少。
AI 中文摘要
乘积公式，又称Trotter公式，是在量子计算机上模拟量子系统最广泛使用且实用的方法之一。本文针对具有分离能标的哈密顿量$H=A+\alpha B$（其中$\alpha$很小），引入两类新的随机化乘积公式。在可分别实现$A$和$B$指数的标准访问模型中，我们的随机化公式实现了$\mathcal O(\alpha^2)$的误差标度，仅需付出将对应确定性公式的门深度加倍的代价；我们进一步证明了确定性乘积公式的$\Omega(\alpha)$下界。在可实现$A+\alpha B_\ell$指数（其中$B = \sum_{\ell}B_\ell$）的更强访问模型中，我们基于Trotter启发式时间动力学资源改进公式（THRIFT）[J. L. Bosse等人，《自然·通讯》16, 2673 (2025)]的随机化公式，实现了$\mathcal O(\alpha^3)$的误差标度，仅需常数因子的预期门开销；我们还在该访问模型中建立了确定性乘积公式的$\Omega(\alpha^2)$下界。数值模拟证实，该方法可减少模拟受物理驱动系统的门数量。
英文摘要
Product formulas, also known as Trotter formulas, are among the most widely used and practical methods for simulating quantum systems on quantum computers. Here we introduce two new classes of randomized product formulas for simulating Hamiltonians with separated energy scales, $H=A+αB$, where $α$ is small. In the standard access model, where one can implement exponentials of $A$ and $B$ separately, our randomized formulas achieve $\mathcal O(α^2)$ error scaling at the cost of only doubling the gate depth of the corresponding deterministic formula. We further prove an $Ω(α)$ lower bound for deterministic product formulas. In a stronger access model, allowing exponentials of $A+αB_\ell$ for $B = \sum_{\ell}B_\ell$, our randomized formula, based on Trotter Heuristic Resource Improved Formulas for Time-dynamics (THRIFT)~[J. L. Bosse et al., Nat. Commun. 16, 2673 (2025)], achieves $\mathcal O(α^3)$ error scaling with only constant-factor expected gate overhead. We also establish an $Ω(α^2)$ lower bound for deterministic product formulas in this access model. Numerical simulations confirm gate-count reductions for simulating physically motivated systems.
