三角形列举与生成子图的量子查询下界

中文摘要 AI 辅助

本文针对两个关系图问题——三角形列举和显式乘法生成子图构造——在一般图查询模型中给出了量子查询下界，在该模型中，量子邻接、度和邻域查询均可在任意叠加态下使用。这两个结果均通过中间多块搜索问题的归约获得。我们扩展了Zhandry（CRYPTO 2019）和Hamoudi与Magniez（ToCT 2023）的量子查询记录框架，以处理双向预言机的记录架构，并给出一个通用的块状可靠性框架，该框架对于任意局部接受投影算子族，给出了它们与有界权重记录子空间的最大重叠的精确算子范数刻画。利用中间搜索问题，我们构造了一个具有Θ(n)个三角形的n顶点图族，在该图族上列出任意常数比例的三角形需要Ω(n^{3/2-o(1)})次量子查询。这是三角形列举的首个非平凡量子下界，该问题由Jiang和Peng（ICML 2026）提出。此外，我们证明，对于每个固定的k≥7，构造一个乘法k-生成子图需要Ω(n^{1+1/(2μ_k)})次量子查询，其中μ_k=k/3+O(1)。对于k∈{7,8}，下界为Ω(n^{5/4})，这与未证明的Erdős围长猜想所暗示的结果一致；对于较大的k，指数1+3/(2k)超过了由Lazebnik、Ustimenko和Woldar（1995）的可证明高围长稠密图所隐含的1+4/(3k)。

英文摘要

This paper gives quantum query lower bounds for two relational graph problems, triangle listing and explicit multiplicative spanner construction, in the general graph query model, where quantum adjacency, degree and neighborhood queries are all available in arbitrary superposition. Both results are obtained by reductions via intermediate multi-block search problems. We extend the quantum query recording framework by Zhandry (CRYPTO 2019) and Hamoudi and Magniez (ToCT 2023) to handle a recording architecture for bidirectional oracles and give a generic blockwise soundness framework, which for arbitrary families of local accepting projectors, gives an exact operator-norm characterization of their maximum overlap with the subspace of bounded weight records. Using the intermediate search problems, we exhibit a family of $n$-vertex graphs with $Θ(n)$ triangles on which listing any constant fraction of the triangles requires $Ω(n^{3/2-o(1)})$ quantum queries. This is the first nontrivial quantum lower bound for triangle listing, a question raised by Jiang and Peng (ICML 2026). Further, we show that, for every fixed $k\ge 7$, constructing a multiplicative $k$-spanner requires $Ω(n^{1+\frac{1}{2μ_k}})$ quantum queries, where $μ_k=k/3+O(1)$. For $k\in\{7,8\}$ the bound is $Ω(n^{5/4})$, which matches what would be implied by an unproven instance of the Erdős girth conjecture, and for large $k$ the exponent $1+\frac{3}{2k}$ exceeds the $1+\frac{4}{3k}$ implied by the provable high girth dense graphs due to Lazebnik, Ustimenko and Woldar, 1995.

发表机构

• National University of Singapore（新加坡国立大学）
• Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）

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