Quantum Property Testing for Bounded-Degree Directed Graphs

Abstract:We study quantum property testing of directed graphs whose maximum in-degree and out-degree are bounded by a fixed constant $d$. For a proximity parameter $\varepsilon$, we prove that every property testable with $O_{\varepsilon,d}(1)$ quantum queries in the bidirectional model, where both incoming and outgoing neighbors are accessible, can also be tested in the quantum unidirectional model, where only outgoing neighbors are accessible, using $n^{1/2-\Omega_{\varepsilon,d}(1)}$ queries. This gives an almost quadratic quantum speedup over the best known generic classical transformation. Our proof has two main ingredients. First, we show that, in the bidirectional model for bounded-degree digraphs, the class of properties testable with constantly many classical queries coincides with the class testable with constantly many quantum queries. Second, we give a transformation from classical bidirectional testers to quantum unidirectional testers by designing a quantum unidirectional algorithm for estimating the frequency vector of constant-radius rooted discs. The algorithm combines quantum counting and Grover search with a correction procedure that removes false local appearances of smaller disc types inside larger neighborhoods. We further show that this transformation is essentially tight. For every sufficiently small fixed $\varepsilon>0$, there is a degree bound $d=d(\varepsilon)$ and an explicit property $P_\varepsilon$ for $d$-bounded-degree digraphs that is $\varepsilon$-testable with $O_\varepsilon(1)$ classical, and hence quantum, bidirectional queries, but requires $ \widetilde{\Omega}_\varepsilon \left(n^{1/2-f'(\varepsilon)}\right) $ quantum queries in the unidirectional model, where $f'(\varepsilon)\to 0$ as $\varepsilon\to 0$.

Quantum Physics (quant-ph); Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS)

arXiv:2604.07954 [quant-ph]

(or arXiv:2604.07954v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2604.07954 Focus to learn more arXiv-issued DOI via DataCite

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