We initiate the study of quantum property testing in sparse directed graphs, and more particularly in the unidirectional model, where the algorithm is allowed to query only the outgoing edges of a vertex. In the classical unidirectional model the problem of testing $k$-star-freeness, and more generally $k$-source-subgraph-freeness, is almost maximally hard for large $k$. We prove that this problem has almost quadratic advantage in the quantum setting. Moreover, we prove that this advantage is nearly tight, by showing a quantum lower bound using the method of dual polynomials on an intermediate problem for a new, property testing version of the $k$-collision problem that was not studied before. To illustrate that not all problems in graph property testing admit such a quantum speedup, we consider the problem of $3$-colorability in the related undirected bounded-degree model, when graphs are now undirected. This problem is maximally hard to test classically, and we show that also quantumly it requires a linear number of queries.

翻译：我们首次研究了稀疏有向图中的量子性质测试问题，特别是在单向模型中，该模型仅允许算法查询顶点的出边。在经典单向模型中，测试$k$-星图自由性（更一般地，$k$-源子图自由性）对于较大的$k$几乎是最大难度的。我们证明了该问题在量子环境下具有几乎二次方的优势。此外，我们通过基于对偶多项式方法对一个中间问题（涉及一种先前未被研究的、性质测试版本的$k$-碰撞问题）给出量子下界，证明了该优势几乎是紧的。为了说明并非所有图性质测试问题都具有此类量子加速，我们考虑了相关无向有界度模型中的$3$-可着色性问题（此时图为无向图）。该问题在经典测试中是最大难度的，我们证明其在量子测试中同样需要线性数量的查询。