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.
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 ε, we show that any property that can be tested with Oε,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 n1/2−Ωε,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 Pε that is ε-testable within Oε(1) classical queries in the bidirectional model, but requires Ω(n1/2−f′(ε)) quantum queries in the unidirectional model, where f′(ε) is a function that approaches 0 as ε 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(n) quantum queries.
In this work, we study bipartiteness and expansion testing, two canonical problems in graph property testing in the bounded-degree model through the lens of quantum query complexity. In the classical setting, it is known that Θ(N) queries are necessary and sufficient for both these testing problems (Goldreich and Ron, 1999, 2000 & 2002), where N denotes the number of vertices of the input graph. Due to their significance, (Ambainis, Childs, and Liu, 2011) initiated the study of these problems in the quantum setting and designed quantum algorithms for bipartiteness and expansion testing that perform O(N1/3) queries, showing a polynomial speedup. They also proved that Ω(N1/4) queries are necessary for expansion testing, but the possibility of an exponential quantum advantage for bipartiteness testing remained open. Despite significant effort, there has been no improvement in these results in the last decade and a half. In this work, we prove essentially tight Ω(N1/3) quantum query lower bounds for both bipartiteness and expansion testing, thereby completely characterizing the quantum query complexity of these problems up to polylogarithmic factors. While our proofs use the polynomial method similarly to Ambainis, Childs, and Liu, we use intermediate problems that we relate to the main problems via reductions, and perform a more precise analysis of the resulting polynomials, leading to the near-optimal lower bounds.
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.
We study the quantum query complexity of several basic graph problems. 1) We prove a tight Θ(n3/2) bound for finding a maximum-out-degree vertex in an n-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 n-vertex undirected graph requires 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)-Cut, which asks for the largest cut with one side of size k. For every constant k≥1, we give an O(n2−1/(k+1))-query quantum algorithm. 4) We establish a reduction to Maximum (k,n−k)-Cut involving k-Graph Collision, which asks whether the marked vertices of a known graph contain a k-clique. For k≥3, this yields unconditional lower bounds with exponents strictly larger than 3/2, showing that this problem is strictly harder than computing maximum degree. Additionally, under a conjecture we propose for k-Graph Collision, the reduction yields a lower bound matching our upper bound above.
We initiate the systematic study of QMA algorithms in the setting of property testing, to which we refer as QMA proofs of proximity (QMAPs). These are quantum query algorithms that receive explicit access to a sublinear-size untrusted proof and are required to accept inputs having a property Π and reject inputs that are ε -far from Π , while only probing a minuscule portion of their input.We investigate the complexity landscape of this model, showing that QMAPs can be exponentially stronger than both classical proofs of proximity and quantum testers. To this end, we extend the methodology of Blais, Brody, and Matulef (Computational Complexity, 2012) to prove quantum property testing lower bounds via reductions from communication complexity. This also resolves a question raised in 2013 by Montanaro and de Wolf (cf. Theory of Computing, 2016).Our algorithmic results include a purpose an algorithmic framework that enables quantum speedups for testing an expressive class of properties, namely, those that are succinctly decomposable . A consequence of this framework is a QMA algorithm to verify the Parity of an n -bit string with O(n2/3) queries and proof length. We also propose a QMA algorithm for testing graph bipartitneness, a property that lies outside of this family, for which there is a quantum speedup.
We study the query complexity of testing quantum juntas. Given an unknown n-qubit unitary or quantum channel, the goal is to decide whether it acts nontrivially on at most k qubits or is ε-far from every such operation. For unitaries, we consider the forward-only model, where the tester has access to U, but not to U† or controlled-U. We give tight bounds for both unitary and channel junta testing. For sufficiently large k, n≥2k, and a broad range of ε, both problems have query complexity Θ(k/(ε2logk)). Our upper bounds are achieved by simple nonadaptive testers using product inputs and single-qubit measurements. For junta channels, this closes the gap by Bao and Yao (COLT 23, TPAMI 2025) between an O(k) upper bound and a Ω(k) lower bound at constant ε, establishing the optimal Θ(k/logk) query complexity. At the heart of both bounds is a strong connection between quantum junta testing and classical support-size testing. For the upper bounds, we reduce quantum junta testing to a generalized support-size problem for distributions over random subsets, and develop a new tester with improved sample complexity. For the lower bounds, we proceed in the reverse direction, reducing classical support-size testing to quantum junta testing through a phase-masked hard ensemble and showing that forward quantum queries can be exactly classicalized on average. Together, these reductions identify support-size testing as the classical core of quantum junta testing and yield matching lower bounds. At constant ε, they also give a separation between Θ(k/logk) forward queries and O(k) queries with inverse access.
Subgraph detection asks whether an n-vertex graph, accessed through queries to its adjacency matrix, contains a copy of a fixed graph H. We prove the first unconditional superlinear lower bounds on the bounded-error quantum query complexity of this problem, answering a longstanding open question. A copy of H is a certificate of constant size, so the adversary method with nonnegative weights cannot prove superlinear lower bounds.
For every fixed r≥4, detecting the clique Kr requires nλr−o(1) queries, where λ4=19/18, the exponents λr increase strictly with r, and λr≥2−42/r+O(1/r). More generally, we prove superlinear lower bounds for detecting every fixed connected graph H with chromatic number c≥4. These bounds approach quadratic as c grows: for sufficiently large c, detection requires n2−O(loglogc/c)−o(1) queries. Chromatic number alone does not characterize the quantum query complexity of subgraph detection: we show that detecting the complete bipartite graph Kr,r requires nβr−o(1) queries, where β10=181/180 and βr≥2−O(1/r).
Our main technical result is a lower bound for finding an all-ones certificate from a known family when the input bits are sampled independently. Its proof combines Zhandry's compressed oracle [CRYPTO 2019] with conditioning on a randomly planted certificate, adapting an argument of Belovs [FOCS 2026]. Our hard instances are built from graphs containing many copies of the desired subgraph with limited overlap. For cliques, we use a construction of Gowers and Janzer [CPC 2021]; for complete bipartite graphs, we use a random construction.