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arXiv:2410.05001 · Leibniz international proceedings in informatics

Quantum Property Testing in Sparse Directed Graphs

Simon Apers, Frédéric Magniez, Sayantan Sen, Dániel Szabó

Published Oct 7, 2024• 0 citations
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Abstract

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 show 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.

DOI
10.4230/lipics.approx/random.2025.32

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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 kkk-star-freeness, and more generally kkk-source-subgraph-freeness, is almost maximally hard for large kkk. We prove that this problem has almost quadratic advantage in the quantum setting. Moreover, we show 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 kkk-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 333-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.
We study quantum property testing of directed graphs whose maximum in-degree and out-degree are bounded by a fixed constant ddd. For a proximity parameter ε\varepsilonε, we prove that every property testable with Oε,d(1)O_{\varepsilon,d}(1)Oε,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 n1/2−Ωε,d(1)n^{1/2-Ω_{\varepsilon,d}(1)}n1/2−Ωε,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 ε>0\varepsilon>0ε>0, there is a degree bound d=d(ε)d=d(\varepsilon)d=d(ε) and an explicit property PεP_\varepsilonPε​ for ddd-bounded-degree digraphs that is ε\varepsilonε-testable with Oε(1)O_\varepsilon(1)Oε​(1) classical, and hence quantum, bidirectional queries, but requires $ \widetildeΩ_\varepsilon \left(n^{1/2-f'(\varepsilon)}\right) $ quantum queries in the unidirectional model, where f′(ε)→0f'(\varepsilon)\to 0f′(ε)→0 as ε→0\varepsilon\to 0ε→0.
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We study the quantum query complexity of several basic graph problems. 1) We prove a tight Θ(n3/2)Θ(n^{3/2})Θ(n3/2) bound for finding a maximum-out-degree vertex in an nnn-vertex tournament, resolving a conjecture of Mande, Paraashar, and Saurabh [FSTTCS'23]. 2) Using the same proof idea, we give a direct proof that finding a maximum-degree vertex in an nnn-vertex undirected graph requires Θ(n3/2)Θ(n^{3/2})Θ(n3/2) quantum queries, a bound implicit in the work of Apers and Lee [CCC'21]. Both lower bounds above hold even for computing only the maximum degree or maximum out-degree. 3) We also study Maximum (k,n−k)(k,n-k)(k,n−k)-Cut, which asks for the largest cut with one side of size kkk. For every constant k≥1k\geq 1k≥1, we give an O(n2−1/(k+1))O(n^{2-1/(k+1)})O(n2−1/(k+1))-query quantum algorithm. 4) We establish a reduction to Maximum (k,n−k)(k,n-k)(k,n−k)-Cut involving kkk-Graph Collision, which asks whether the marked vertices of a known graph contain a kkk-clique. For k≥3k\geq 3k≥3, this yields unconditional lower bounds with exponents strictly larger than 3/23/23/2, showing that this problem is strictly harder than computing maximum degree. Additionally, under a conjecture we propose for kkk-Graph Collision, the reduction yields a lower bound matching our upper bound above.
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