The edge list model is arguably the simplest input model for graphs, where the graph is specified by a list of its edges. In this model, we study the quantum query complexity of three variants of the triangle finding problem. The first asks whether there exists a triangle containing a target edge and raises general questions about the hiding of a problem's input among irrelevant data. The second asks whether there exists a triangle containing a target vertex and raises general questions about the shuffling of a problem's input. The third asks whether there exists a triangle; this problem bridges the $3$-distinctness and $3$-sum problems, which have been extensively studied by both cryptographers and complexity theorists. We provide tight or nearly tight results for these problems as well as some first answers to the general questions they raise. Furthermore, given any graph with low maximum degree, such as a typical random sparse graph, we prove that the quantum query complexity of finding a length-$k$ cycle in its length-$m$ edge list is $m^{3/4-1/(2^{k+2}-4)\pm o(1)}$, which matches the best-known upper bound for the quantum query complexity of $k$-distinctness on length-$m$ inputs up to an $m^{o(1)}$ factor. We prove the lower bound by developing new techniques within Zhandry's recording query framework [CRYPTO '19] as generalized by Hamoudi and Magniez [ToCT '23]. These techniques extend the framework to treat any non-product distribution that results from conditioning a product distribution on the absence of rare events. We prove the upper bound by adapting Belovs's learning graph algorithm for $k$-distinctness [FOCS '12]. Finally, assuming a plausible conjecture concerning only cycle finding, we show that the lower bound can be lifted to an essentially tight lower bound on the quantum query complexity of $k$-distinctness, which is a long-standing open question.

翻译：边列表模型可以说是最简单的图输入模型，其中图由其边的列表指定。在此模型中，我们研究了三角形查找问题的三个变体的量子查询复杂度。第一个问题询问是否存在包含目标边的三角形，并引发了关于问题输入在无关数据中隐藏的一般性问题。第二个问题询问是否存在包含目标顶点的三角形，并引发了关于问题输入洗牌的一般性问题。第三个问题询问是否存在三角形；该问题连接了已被密码学家和复杂性理论家广泛研究的$3$-相异性和$3$-和问题。我们为这些问题提供了紧或近乎紧的结果，并对它们引发的一般性问题给出了初步解答。此外，对于任何具有低最大度的图（例如典型的随机稀疏图），我们证明了在其长度为$m$的边列表中查找长度为$k$的环的量子查询复杂度为$m^{3/4-1/(2^{k+2}-4)\pm o(1)}$，这与长度为$m$的输入上$k$-相异性的量子查询复杂度已知最佳上界匹配，相差一个$m^{o(1)}$因子。我们通过开发Zhandry记录查询框架[CRYPTO '19]（由Hamoudi和Magniez推广[ToCT '23]）内的新技术证明了该下界。这些技术扩展了该框架，以处理由在稀有事件缺失条件下对乘积分布进行条件化所产生的任何非乘积分布。我们通过改编Belovs针对$k$-相异性的学习图算法[FOCS '12]证明了上界。最后，假设一个仅关于环查找的合理猜想，我们表明该下界可以被提升为$k$-相异性量子查询复杂度的本质紧下界，这是一个长期存在的开放问题。