Quantum query complexity of minor-closed graph properties

. To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in

o(n^{3/2})

queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

Related papers

Quantum Query Complexity of Minor-Closed Graph Properties

Andrew M. Childs, Robin Kothari

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$ -vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity $\Theta(n^{3/2})$ . To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in $o(n^{3/2})$ queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity $\Theta(n^{3/2})$. To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in $o(n^{3/2})$ queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

SIAM Journal on ComputingJan 1, 201223 citations

Quantum algorithms for path and cycle containment problems

Arjan Cornelissen, Amin Shiraz Gilani, Subhasree Patro

The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph $H$ is present in an input graph $G$ , 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 $k$ . 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 $k$ , or at most $k$ . 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 $\widetilde{O}(n^{3/2-α_k})$ , where $α_k \in Θ(c^{-k})$ and $c = \sqrt{3+\sqrt{17}}/2 \approx 1.33$ , beating the previous best upper bound $O(n^{3/2})$ 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 $O(\sqrt{n})$ query algorithm.The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph $H$ is present in an input graph $G$, 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 $k$. 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 $k$, or at most $k$. 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 $\widetilde{O}(n^{3/2-α_k})$, where $α_k \in Θ(c^{-k})$ and $c = \sqrt{3+\sqrt{17}}/2 \approx 1.33$, beating the previous best upper bound $O(n^{3/2})$ 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 $O(\sqrt{n})$ query algorithm.

quant-pharXiv:2605.09017Leibniz international proceedings in informaticsMay 12, 20260 citations

Quantum query complexity of minor-closed graph properties

Andrew M. Childs, Robin Kothari

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$ -vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Θ(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Θ(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

arXiv:1011.1443Leibniz international proceedings in informaticsNov 5, 20103 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.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:2609.40263Oct 1, 20260 citations

Efficient algorithms in quantum query complexity

Robin Kothari

In this thesis we provide new upper and lower bounds on the quantum query complexity of a diverse set of problems. Specifically, we study quantum algorithms for Hamiltonian simulation, matrix multiplication, oracle identification, and graph-property recognition. \n \nFor the Hamiltonian simulation problem, we provide a quantum algorithm with query complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Our algorithm is based on a new quantum algorithm for implementing unitary matrices that can be written as linear combinations of efficiently implementable unitary gates. This algorithm uses a new form of ``oblivious amplitude amplification'' that can be applied even though the reflection about the input state is unavailable. \n \nIn the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set of size M, and our task is to identify x. We present the first quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered gamma_2-norm semidefinite program. \n \nWe then study the quantum query complexity of matrix multiplication and related problems over rings, semirings, and the Boolean semiring in particular. Our main result is an output-sensitive algorithm for Boolean matrix multiplication that multiplies two n x n Boolean matrices with query complexity O(n sqrt{l}), where l is the sparsity of the output matrix. The algorithm is based on a reduction to the graph collision problem and a new algorithm for graph collision. \n \nFinally, we study the quantum query complexity of minor-closed graph properties and show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Theta(n^{3/2}) and those that do have such a characterization can be solved strictly faster, with o(n^{3/2}) queries. Our lower bound is based on a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.In this thesis we provide new upper and lower bounds on the quantum query complexity of a diverse set of problems. Specifically, we study quantum algorithms for Hamiltonian simulation, matrix multiplication, oracle identification, and graph-property recognition. \n \nFor the Hamiltonian simulation problem, we provide a quantum algorithm with query complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Our algorithm is based on a new quantum algorithm for implementing unitary matrices that can be written as linear combinations of efficiently implementable unitary gates. This algorithm uses a new form of ``oblivious amplitude amplification'' that can be applied even though the reflection about the input state is unavailable. \n \nIn the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set of size M, and our task is to identify x. We present the first quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered gamma_2-norm semidefinite program. \n \nWe then study the quantum query complexity of matrix multiplication and related problems over rings, semirings, and the Boolean semiring in particular. Our main result is an output-sensitive algorithm for Boolean matrix multiplication that multiplies two n x n Boolean matrices with query complexity O(n sqrt{l}), where l is the sparsity of the output matrix. The algorithm is based on a reduction to the graph collision problem and a new algorithm for graph collision. \n \nFinally, we study the quantum query complexity of minor-closed graph properties and show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Theta(n^{3/2}) and those that do have such a characterization can be solved strictly faster, with o(n^{3/2}) queries. Our lower bound is based on a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

Aug 13, 201420 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.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:2610.13906Oct 12, 20260 citations

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an

n

\Theta(n^{3/2})

o(n^{3/2})

queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an

n

\Theta(n^{3/2})

o(n^{3/2})

queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph

H

is present in an input graph

G

, 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

k

. 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

k

, or at most

k

. 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

\widetilde{O}(n^{3/2-α_k})

, where

α_k \in Θ(c^{-k})

and

c = \sqrt{3+\sqrt{17}}/2 \approx 1.33

, beating the previous best upper bound

O(n^{3/2})

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(\sqrt{n})

query algorithm.

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an

n

-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Θ(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

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.

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.