边着色格图
Edge coloring lattice graphs
AI总结：
本文发展了无限格图的边着色理论，提出了基于平移诱导条件的近似最小与最小边着色方法，并证明其在量子模拟、优化及态验证中可提供最小深度量子电路。
摘要：
我们发展了无限格图边着色理论，证明了格图局部补丁的正常边着色通过平移诱导整个格图正常边着色的充分必要条件。该条件构成了寻找无限格图近似最小或最小边着色方法的基石。在要求近似最小边着色时，运行时间为$O(μ^2 D^4)$，其中$μ$是格图一个单元（或“基图”）内的边数，$D$是存在单元间边的两单元的最大距离。在要求最小边着色时，我们缺乏运行时间的上界，但发现这在实践中并不构成限制；我们利用该方法对$k\leq 6$的平面所有$k$-均匀铺砌的网格进行了最小边着色，仅使用适度的计算资源。我们发现所有这些格图均属于Vizing class~I。将边着色与量子电路相关联，我们的工作通过在量子模拟、量子优化和量子态验证领域提供最小深度量子电路，找到了直接应用。
英文摘要：
We develop the theory of the edge coloring of infinite lattice graphs, proving a necessary and sufficient condition for a proper edge coloring of a patch of a lattice graph to induce a proper edge coloring of the entire lattice graph by translation. This condition forms the cornerstone of a method that finds nearly minimal or minimal edge colorings of infinite lattice graphs. In case a nearly minimal edge coloring is requested, the running time is $O(μ^2 D^4)$, where $μ$ is the number of edges in one cell (or `basis graph') of the lattice graph and $D$ is the maximum distance between two cells so that there is an edge from within one cell to the other. In case a minimal edge coloring is requested, we lack an upper bound on the running time, which we find need not pose a limitation in practice; we use the method to minimal edge color the meshes of all $k$-uniform tilings of the plane for $k\leq 6$, while utilizing modest computational resources. We find that all these lattice graphs are Vizing class~I. Relating edge colorings to quantum circuits, our work finds direct application by offering minimal-depth quantum circuits in the areas of quantum simulation, quantum optimization, and quantum state verification.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
