Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.01752

Near-optimal quantum query lower bounds on bipartiteness and expansion testing in the bounded-degree graph model

Chandrima Kayal, Sayantan Sen, Dániel Szabó

quant-phcs.DSSubmitted Oct 1, 2026• Announced Oct 2, 2026• 0 citations
arXivHTMLPDF

Abstract

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 $\widetildeΘ(\sqrt{N})$ 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 $\widetilde{O}(N^{1/3})$ queries, showing a polynomial speedup. They also proved that $\widetildeΩ(N^{1/4})$ 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 $\widetildeΩ(N^{1/3})$ 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.

Comments
38 pages

Related papers

(Quantum) Complexity of Testing Signed Graph Clusterability

Chen, Kuo-Chin, Simon Apers, Min-Hsiu Hsieh

This study examines clusterability testing for a signed graph in the bounded-degree model. Our contributions are two-fold. First, we provide a quantum algorithm with query complexity $\tilde{O}(N^{1/3})$ for testing clusterability, which yields a polynomial speedup over the best classical clusterability tester known [arXiv:2102.07587]. Second, we prove an $\tildeΩ(\sqrt{N})$ classical query lower bound for testing clusterability, which nearly matches the upper bound from [arXiv:2102.07587]. This settles the classical query complexity of clusterability testing, and it shows that our quantum algorithm has an advantage over any classical algorithm.

arXiv:2311.10480•Leibniz international proceedings in informatics•Nov 17, 2023•0 citations

Superlinear Quantum Query Lower Bounds for Subgraph Detection

Amin Shiraz Gilani, Xingyu Zhou

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 $r\ge 4$, detecting the clique $K_r$ requires $n^{λ_r-o(1)}$ queries, where $λ_4=19/18$, the exponents $λ_r$ increase strictly with $r$, and $λ_r\ge 2-4\sqrt{2/r}+O(1/r)$. More generally, we prove superlinear lower bounds for detecting every fixed connected graph $H$ with chromatic number $c\ge 4$. These bounds approach quadratic as $c$ grows: for sufficiently large $c$, detection requires $n^{2-O(\sqrt{\log\log c/c})-o(1)}$ queries. Chromatic number alone does not characterize the quantum query complexity of subgraph detection: we show that detecting the complete bipartite graph $K_{r,r}$ requires $n^{β_r-o(1)}$ queries, where $β_{10}=181/180$ and $β_r\ge 2-O(1/\sqrt{r})$. 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.

quant-pharXiv:2609.40263•Oct 1, 2026•0 citations

Quantum Property Testing in Sparse Directed Graphs

Simon Apers, Frédéric Magniez, Sayantan Sen +1 more

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.

arXiv:2410.05001•Leibniz international proceedings in informatics•Oct 7, 2024•0 citations

Quantum Query Complexity of Maximum Degree and its Generalizations

Amin Shiraz Gilani, Nikhil S. Mande, Subhasree Patro +1 more

We study the quantum query complexity of several basic graph problems. 1) We prove a tight $Θ(n^{3/2})$ 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 $Θ(n^{3/2})$ 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 $k\geq 1$, we give an $O(n^{2-1/(k+1)})$-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 $k\geq 3$, 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.

quant-pharXiv:2610.13906•Oct 12, 2026•0 citations

Quantum Property Testing for Bounded-Degree Directed Graphs

Pan Peng, Jingyu Wu

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 $n^{1/2-Ω_{\varepsilon,d}(1)}$ 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 $ \widetildeΩ_\varepsilon \left(n^{1/2-f'(\varepsilon)}\right) $ quantum queries in the unidirectional model, where $f'(\varepsilon)\to 0$ as $\varepsilon\to 0$.

quant-pharXiv:2604.07954•Apr 10, 2026•0 citations

Improved Quantum Query Bounds for Boolean Matrix Product Verification

Amin Shiraz Gilani, François Le Gall, Xingyu Zhou

We prove the first non-trivial upper bound for the quantum query complexity of Boolean Matrix Product Verification ($\mathsf{BMPV}$), answering a longstanding open question in quantum query complexity. For $n\times n$ matrices, our upper bound is $\widetilde O(n^{17/12})$, improving on the standard $O(n^{3/2})$ bound obtained using Grover search by Buhrman and Špalek [SODA 2006]. We complement this result by showing an $Ω(n^{5/4})$ lower bound, which improves over the previous best known lower bound of $\widetildeΩ(n^{19/18})$ by Childs, Kimmel, and Kothari [ESA 2012]. Our approach centers on a connection with Orthogonal Vectors ($\mathsf{OV}$), which asks whether an indexed list of $n$ Boolean vectors of dimension $n$ contains two vectors with disjoint supports. In particular, we prove equivalences between $\mathsf{OV}$ and $\mathsf{BMPV}$ and establish the above bounds for $\mathsf{OV}$. We also prove a tight $\widetilde Θ(n^{3/2})$ bound for a variant of $\mathsf{BMPV}$ that asks whether the product contains a given row vector. Together, these results imply a polynomial separation between the quantum query complexities of deciding whether a graph has radius at most two and whether it has diameter at most two.

quant-pharXiv:2609.40180•Oct 1, 2026•0 citations
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)\widetildeΘ(\sqrt{N})Θ(N​) queries are necessary and sufficient for both these testing problems (Goldreich and Ron, 1999, 2000 & 2002), where NNN 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)\widetilde{O}(N^{1/3})O(N1/3) queries, showing a polynomial speedup. They also proved that Ω~(N1/4)\widetildeΩ(N^{1/4})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)\widetildeΩ(N^{1/3})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.
This study examines clusterability testing for a signed graph in the bounded-degree model. Our contributions are two-fold. First, we provide a quantum algorithm with query complexity O~(N1/3)\tilde{O}(N^{1/3})O~(N1/3) for testing clusterability, which yields a polynomial speedup over the best classical clusterability tester known [arXiv:2102.07587]. Second, we prove an Ω~(N)\tildeΩ(\sqrt{N})Ω~(N​) classical query lower bound for testing clusterability, which nearly matches the upper bound from [arXiv:2102.07587]. This settles the classical query complexity of clusterability testing, and it shows that our quantum algorithm has an advantage over any classical algorithm.
Subgraph detection asks whether an nnn-vertex graph, accessed through queries to its adjacency matrix, contains a copy of a fixed graph HHH. 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 HHH is a certificate of constant size, so the adversary method with nonnegative weights cannot prove superlinear lower bounds. For every fixed r≥4r\ge 4r≥4, detecting the clique KrK_rKr​ requires nλr−o(1)n^{λ_r-o(1)}nλr​−o(1) queries, where λ4=19/18λ_4=19/18λ4​=19/18, the exponents λrλ_rλr​ increase strictly with rrr, and λr≥2−42/r+O(1/r)λ_r\ge 2-4\sqrt{2/r}+O(1/r)λr​≥2−42/r​+O(1/r). More generally, we prove superlinear lower bounds for detecting every fixed connected graph HHH with chromatic number c≥4c\ge 4c≥4. These bounds approach quadratic as ccc grows: for sufficiently large ccc, detection requires n2−O(log⁡log⁡c/c)−o(1)n^{2-O(\sqrt{\log\log c/c})-o(1)}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,rK_{r,r}Kr,r​ requires nβr−o(1)n^{β_r-o(1)}nβr​−o(1) queries, where β10=181/180β_{10}=181/180β10​=181/180 and βr≥2−O(1/r)β_r\ge 2-O(1/\sqrt{r})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.
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 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.
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.
We prove the first non-trivial upper bound for the quantum query complexity of Boolean Matrix Product Verification (BMPV\mathsf{BMPV}BMPV), answering a longstanding open question in quantum query complexity. For n×nn\times nn×n matrices, our upper bound is O~(n17/12)\widetilde O(n^{17/12})O(n17/12), improving on the standard O(n3/2)O(n^{3/2})O(n3/2) bound obtained using Grover search by Buhrman and Špalek [SODA 2006]. We complement this result by showing an Ω(n5/4)Ω(n^{5/4})Ω(n5/4) lower bound, which improves over the previous best known lower bound of Ω~(n19/18)\widetildeΩ(n^{19/18})Ω(n19/18) by Childs, Kimmel, and Kothari [ESA 2012]. Our approach centers on a connection with Orthogonal Vectors (OV\mathsf{OV}OV), which asks whether an indexed list of nnn Boolean vectors of dimension nnn contains two vectors with disjoint supports. In particular, we prove equivalences between OV\mathsf{OV}OV and BMPV\mathsf{BMPV}BMPV and establish the above bounds for OV\mathsf{OV}OV. We also prove a tight Θ~(n3/2)\widetilde Θ(n^{3/2})Θ(n3/2) bound for a variant of BMPV\mathsf{BMPV}BMPV that asks whether the product contains a given row vector. Together, these results imply a polynomial separation between the quantum query complexities of deciding whether a graph has radius at most two and whether it has diameter at most two.