Understanding algorithmic error accumulation in quantum simulation is crucial due to its fundamental significance and practical applications in simulating quantum many-body system dynamics. Conventional theories typically apply the triangle inequality to provide an upper bound for the error. However, these often yield overly conservative and inaccurate estimates as they neglect error interference -- a phenomenon where errors in different segments can destructively interfere. Here, we introduce a novel method that directly estimates the long-time algorithmic errors with multiple segments, thereby establishing a comprehensive framework for characterizing algorithmic error interference. We identify the sufficient and necessary condition for strict error interference and introduce the concept of approximate error interference, which is more broadly applicable to scenarios such as power-law interaction models, the Fermi-Hubbard model, and higher-order Trotter formulas. Our work demonstrates significant improvements over prior ones and opens new avenues for error analysis in quantum simulation, offering potential advancements in both theoretical algorithm design and experimental implementation of Hamiltonian simulation.

翻译：理解量子模拟中的算法误差累积至关重要，这源于其在模拟量子多体系统动力学方面的基础意义和实际应用。传统理论通常应用三角不等式来提供误差的上界。然而，由于它们忽略了误差干涉——即不同区段的误差可能发生相消干涉的现象，这些理论往往给出过于保守且不准确的估计。在此，我们引入一种新方法，直接估计包含多个区段的长时间算法误差，从而建立了一个用于表征算法误差干涉的综合性框架。我们确定了严格误差干涉的充分必要条件，并引入了近似误差干涉的概念，该概念更广泛地适用于幂律相互作用模型、费米-哈伯德模型以及高阶Trotter公式等场景。我们的工作展示了相较于先前研究的显著改进，并为量子模拟中的误差分析开辟了新途径，有望在哈密顿量模拟的理论算法设计和实验实现方面带来潜在进展。