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arXiv:2610.13906

Quantum Query Complexity of Maximum Degree and its Generalizations

Amin Shiraz Gilani, Nikhil S. Mande, Subhasree Patro, Nithish Raja

quant-phcs.CCSubmitted Oct 9, 2026• Announced Oct 12, 2026• 0 citations
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Abstract

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.

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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.

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arXiv:2408.03864•Leibniz international proceedings in informatics•Jan 1, 2024•0 citations

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

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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.

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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
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.
A tournament solution records a set of winners from a tournament, that is, from the outcomes of all pairwise comparisons among NNN alternatives. We study the quantum query complexity of computing tournament solutions when the outcome of each comparison has to be learned by a query. Classically, the picture is uniform: the Copeland set, the uncovered set and the top cycle all require Θ(N2)Θ(N^2)Θ(N2) queries, and so does finding a single vertex of maximum out-degree. We show that the quantum query complexity of the top cycle can be computed with Θ~(N)\tildeΘ(N)Θ~(N) queries, and a Condorcet winner can be found with Θ~(N)\tilde Θ(\sqrt N)Θ~(N​) queries. Finding a strong king requires Θ(N3/2)Θ(N^{3/2})Θ(N3/2) queries, the same as finding a vertex of maximum out-degree. We also give a self-contained alternative proof of the known Θ(N3/2)Θ(N^{3/2})Θ(N3/2) bound for finding a vertex of maximum degree, established by Gilani, Mande, Patro, and Raja. Our lower bounds reduce counting problems to tournament problems using Ambainis's adversary method. For the top cycle, we find a Hamiltonian path and locate the prefix forming the top cycle by divide and conquer.
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.
The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph HHH is present in an input graph GGG, has been the subject of considerable study. However, even for relatively simple subgraphs, such as paths and cycles, a complete understanding of their query complexities remains elusive. In this work, we consider several variants of path- and cycle-containment problems in the adjacency matrix model, where we search for paths or cycles of constant length kkk. We compare the settings where the graphs are directed or undirected, where the goal is to detect or find the existence of a path/cycle, and where the path/cycle we are looking for has length exactly kkk, or at most kkk. We also consider several promise versions of these problems, where we suppose that the input graph has a certain structure. We characterize the relative difficulty of these variants of the path/cycle-containment problems, by relating them to one another using randomized reductions, and grouping them into equivalence classes. When we restrict our attention to path-containment problems, we get a dichotomy result. Some of the path-containment problems can be solved using a linear number of queries, and all the others are equivalent to one another (and additionally to several cycle-containment problems) under randomized reductions, up to constant overhead. For the latter equivalence class, we prove a novel quantum-walk-based algorithm that achieves query complexity O~(n3/2−αk)\widetilde{O}(n^{3/2-α_k})O(n3/2−αk​), where αk∈Θ(c−k)α_k \in Θ(c^{-k})αk​∈Θ(c−k) and c=3+17/2≈1.33c = \sqrt{3+\sqrt{17}}/2 \approx 1.33c=3+17​​/2≈1.33, beating the previous best upper bound O(n3/2)O(n^{3/2})O(n3/2) on its query complexity. We also provide a conditional lower bound based on the graph-collision problem, which implies that this equivalence class does not admit linear-query quantum algorithms unless graph collision admits an O(n)O(\sqrt{n})O(n​) query algorithm.
In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.
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.
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.