发表机构
University of Maryland（马里兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了子图检测问题（通过邻接矩阵查询判断图是否包含固定子图）的首个无条件超线性量子查询下界，针对团和完全二分图给出了具体指数下界，并引入了基于压缩预言机和随机植入证书的新技术。
AI 中文摘要
子图检测问题询问一个通过邻接矩阵查询访问的 $n$ 顶点图是否包含固定图 $H$ 的副本。我们证明了该问题有界误差量子查询复杂性的第一个无条件超线性下界，回答了一个长期悬而未决的问题。$H$ 的副本是一个恒定大小的证书，因此具有非负权重的对手方法无法证明超线性下界。对于每个固定的 $r\ge 4$，检测团 $K_r$ 需要 $n^{\lambda_r-o(1)}$ 次查询，其中 $\lambda_4=19/18$，指数 $\lambda_r$ 随 $r$ 严格增加，且 $\lambda_r\ge 2-4\sqrt{2/r}+O(1/r)$。更一般地，我们证明了检测每个具有色数 $c\ge 4$ 的固定连通图 $H$ 的超线性下界。这些下界随着 $c$ 的增长而接近二次：对于足够大的 $c$，检测需要 $n^{2-O(\sqrt{\log\log c/c})-o(1)}$ 次查询。仅凭色数并不能刻画子图检测的量子查询复杂性：我们证明检测完全二分图 $K_{r,r}$ 需要 $n^{\beta_r-o(1)}$ 次查询，其中 $\beta_{10}=181/180$ 且 $\beta_r\ge 2-O(1/\sqrt{r})$。我们的主要技术结果是，当输入位独立采样时，从已知族中找到全一证书的下界。其证明将 Zhandry 的压缩预言机（CRYPTO 2019）与对随机植入证书的条件化相结合，改编了 Belovs（FOCS 2026）的一个论证。我们的困难实例由包含许多所需子图副本且重叠有限的图构建。对于团，我们使用 Gowers 和 Janzer（CPC 2021）的构造；对于完全二分图，我们使用随机构造。
英文摘要
Subgraph detection asks whether an $n$-vertex graph, accessed through queries to its adjacency matrix, contains a copy of a fixed graph $H$. We prove the first unconditional superlinear lower bounds on the bounded-error quantum query complexity of this problem, answering a longstanding open question. A copy of $H$ is a certificate of constant size, so the adversary method with nonnegative weights cannot prove superlinear lower bounds. For every fixed $r\ge 4$, detecting the clique $K_r$ requires $n^{λ_r-o(1)}$ queries, where $λ_4=19/18$, the exponents $λ_r$ increase strictly with $r$, and $λ_r\ge 2-4\sqrt{2/r}+O(1/r)$. More generally, we prove superlinear lower bounds for detecting every fixed connected graph $H$ with chromatic number $c\ge 4$. These bounds approach quadratic as $c$ grows: for sufficiently large $c$, detection requires $n^{2-O(\sqrt{\log\log c/c})-o(1)}$ queries. Chromatic number alone does not characterize the quantum query complexity of subgraph detection: we show that detecting the complete bipartite graph $K_{r,r}$ requires $n^{β_r-o(1)}$ queries, where $β_{10}=181/180$ and $β_r\ge 2-O(1/\sqrt{r})$. Our main technical result is a lower bound for finding an all-ones certificate from a known family when the input bits are sampled independently. Its proof combines Zhandry's compressed oracle [CRYPTO 2019] with conditioning on a randomly planted certificate, adapting an argument of Belovs [FOCS 2026]. Our hard instances are built from graphs containing many copies of the desired subgraph with limited overlap. For cliques, we use a construction of Gowers and Janzer [CPC 2021]; for complete bipartite graphs, we use a random construction.
Comments41 pages; v2: minor revisions and improvements
