← 研踪 PaperTrailarXiv:2611.06751 · v1
A Quantum Lower Bound for Triangle Finding
首次提交 2026-11-05 00:27
版本更新 2026-11-05 00:27
雷达观测 2026-11-05 12:13
摘要
Triangle finding is a fundamental problem in graph theory that has played a central role in the development of quantum algorithmic techniques. For $N$-vertex graphs given by their adjacency matrices, a long line of work has improved the quantum query upper bound from $O(N^{3/2})$ to $O(N^{5/4})$. However, the best known lower bound has remained the elementary $Ω(N)$ bound obtained from unstructured search. We prove the first superlinear quantum query lower bound for triangle finding, showing that $Ω(N^{13/12})$ queries are necessary. Our result also implies the same lower bound for triangle detection and for detecting every fixed connected non-bipartite graph. The proof builds on the compressed oracle approach of Zhandry and the transfer approaches of Belovs. It compares the computation on graphs with a planted triangle to the computation on random graphs conditioned to be triangle-free, through a map on the input registers that almost commutes with every query. We average over local changes to the input while preserving its distribution, through a procedure known as conditional resampling. Our main technical contribution is a set of estimates that let us control the query error under this averaging, despite the dependence introduced by conditioning a dense graph to be triangle-free.
作者备注：37 pages
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v1 · 2026-11-05 00:27 · 最新观测版本
A Quantum Lower Bound for Triangle Finding
Triangle finding is a fundamental problem in graph theory that has played a central role in the development of quantum algorithmic techniques. For $N$-vertex graphs given by their adjacency matrices, a long line of work has improved the quantum query upper bound from $O(N^{3/2})$ to $O(N^{5/4})$. However, the best known lower bound has remained the elementary $Ω(N)$ bound obtained from unstructured search. We prove the first superlinear quantum query lower bound for triangle finding, showing that $Ω(N^{13/12})$ queries are necessary. Our result also implies the same lower bound for triangle detection and for detecting every fixed connected non-bipartite graph. The proof builds on the compressed oracle approach of Zhandry and the transfer approaches of Belovs. It compares the computation on graphs with a planted triangle to the computation on random graphs conditioned to be triangle-free, through a map on the input registers that almost commutes with every query. We average over local changes to the input while preserving its distribution, through a procedure known as conditional resampling. Our main technical contribution is a set of estimates that let us control the query error under this averaging, despite the dependence introduced by conditioning a dense graph to be triangle-free.
查看此版本解读 →查看此版本原文 ↗