Quantum simulation is widely regarded as one of the most promising applications of quantum computing. A critical challenge in this domain is understanding and quantifying the accumulation of algorithmic errors over time, which is essential for designing more efficient simulation algorithms and for assessing the resources required to achieve quantum advantage. Conventional error analyses typically rely on the triangle inequality to bound the total simulation error, but such approaches tend to overestimate errors by ignoring error interference—a phenomenon in which errors from different simulation segments partially cancel. Here, we introduce a new framework for directly estimating long-time algorithmic errors in segmented quantum simulations. Our approach captures the full structure of error interference, enabling significantly tighter and more accurate error bounds. We identify both necessary and sufficient conditions for strict error interference and propose the notion of approximate error interference to account for realistic, imperfect cancellation. We demonstrate the broad applicability of our framework across a range of models and settings, including Heisenberg and Fermi-Hubbard systems, lattice Hamiltonians with power-law interactions, higher-order Trotter decompositions, and adiabatic evolution. By providing a unified and practical methodology for analyzing error interference, our Letter advances the theoretical understanding of quantum simulation and informs the design and benchmarking of algorithms for near-term and future quantum hardware.

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• Quantum algorithms
• Quantum simulation