Trotter误差的排序感知理论
Ordering-Aware Theory of Trotter Error
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中文总结 AI 辅助
本文系统研究乘积公式中哈密顿量项排序对模拟精度的影响，提出排序敏感的Trotter误差界，并利用局域结构将排序优化简化为图问题，证明顺序排序优于砖墙电路，确立排序为提升模拟精度的重要自由度。
中文摘要 AI 辅助
乘积公式是模拟量子多体动力学的基本工具，但其应用中的一个基本自由度——哈密顿量项的排序——在很大程度上仍未得到探索。在此，我们系统地研究此类排序效应，量化其对模拟精度的影响，并利用它们设计更精确的乘积公式模拟。我们为几何局域哈密顿量开发了一个可高效计算、对排序敏感的有限步长Trotter误差界，适用于任意高阶乘积公式。对于平移不变的一维链，我们的界在小规模测试中接近精确的二阶、四阶和六阶Trotter误差，而在大规模系统上，它们比之前的高阶解析界改善了数个数量级。我们进一步利用误差界和估计的局域结构来高效优化排序，将长链问题简化为加权de Bruijn图上的最小平均权重环问题。对于随机平移不变链，顺序（阶梯状）排序始终优于标准砖墙电路；当电路深度受限时，优化器转而青睐波状并行顺序排序，其保留了相对于砖墙电路的大部分精度优势。这些结果确立了排序作为理解和改进乘积公式模拟的一个有用自由度。
英文摘要
Product formulas are a basic tool for simulating quantum many-body dynamics, but a fundamental degree of freedom in their application---the ordering of Hamiltonian terms---has remained largely unexplored. Here we systematically study such ordering effects, quantifying their impact on simulation accuracy and exploiting them to design more accurate product-formula simulations. We develop an efficiently computable, ordering-sensitive bound on the finite-step-size Trotter error for geometrically local Hamiltonians for arbitrarily high-order product formulas. For translation-invariant one-dimensional chains, our bounds are close to exact second-, fourth-, and sixth-order Trotter errors in small-system tests, while on large-scale systems they improve on previous higher-order analytical bounds by orders of magnitude. We further use the local structure of the error bounds and estimates to optimize ordering efficiently, reducing the long-chain problem to a minimum-mean-weight-cycle problem on a weighted de Bruijn graph. For random translation-invariant chains, sequential (staircase-like) orderings consistently outperform standard brickwall circuits; when circuit depth is restricted, the optimizer instead favors wave-like parallel-sequential orderings, which preserve much of the accuracy advantage over brickwall circuits. These results establish ordering as a useful degree of freedom for both understanding and improving product-formula simulation.
发表机构
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（量子信息与计算联合中心，美国国家标准与技术研究院/马里兰大学）
- Department of Computer Science, University of Maryland（马里兰大学计算机科学系）
- Joint Quantum Institute, NIST/University of Maryland（量子信息联合研究所，美国国家标准与技术研究院/马里兰大学）
- Quantinuum
- Institute for Advanced Computer Studies, University of Maryland（马里兰大学高级计算机研究所）
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