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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. |
