We study quantum property testing for directed graphs with maximum in-degree and out-degree bounded by some universal constant $d$. For a proximity parameter $\varepsilon$, we show that any property that can be tested with $O_{\varepsilon,d}(1)$ queries in the classical bidirectional model, where both incoming and outgoing edges are accessible, can also be tested in the quantum unidirectional model, where only outgoing edges are accessible, using $n^{1/2 - Ω_{\varepsilon,d}(1)}$ queries. This yields an almost quadratic quantum speedup over the best known classical algorithms in the unidirectional model. Moreover, we prove that our transformation is almost tight by giving an explicit property $P_\varepsilon$ that is $\varepsilon$-testable within $O_\varepsilon(1)$ classical queries in the bidirectional model, but requires $\widetildeΩ(n^{1/2-f'(\varepsilon)})$ quantum queries in the unidirectional model, where $f'(\varepsilon)$ is a function that approaches $0$ as $\varepsilon$ approaches $0$. As a byproduct, we show that in the unidirectional model, the number of occurrences of any constant-size subgraph $H$ can be approximated up to additive error $δn$ using $o(\sqrt{n})$ quantum queries.

翻译：我们研究最大入度和出度由某个通用常数 $d$ 有界的有向图的量子性质检测。对于邻近参数 $\varepsilon$，我们证明，在经典双向模型（即可同时访问入边和出边）中可用 $O_{\varepsilon,d}(1)$ 次查询检测的任何性质，在量子单向模型（即仅可访问出边）中也可用 $n^{1/2 - Ω_{\varepsilon,d}(1)}$ 次查询检测。这相较于已知最佳经典算法在单向模型中实现了近乎二次的量子加速。此外，我们通过给出一个显式性质 $P_\varepsilon$ 证明我们的变换几乎是紧的：该性质在双向模型中可用 $O_\varepsilon(1)$ 次经典查询进行 $\varepsilon$ 检测，但在单向模型中需要 $\widetildeΩ(n^{1/2-f'(\varepsilon)})$ 次量子查询，其中 $f'(\varepsilon)$ 是一个在 $\varepsilon$ 趋近于 $0$ 时趋向于 $0$ 的函数。作为副产品，我们证明在单向模型中，任意常数大小子图 $H$ 的出现次数可用 $o(\sqrt{n})$ 次量子查询近似到加法误差 $δn$ 内。