A tournament solution records a set of winners from a tournament, that is, from the outcomes of all pairwise comparisons among $N$ 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 $Θ(N^2)$ 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 $\tildeΘ(N)$ queries, and a Condorcet winner can be found with $\tilde Θ(\sqrt N)$ queries. Finding a strong king requires $Θ(N^{3/2})$ queries, the same as finding a vertex of maximum out-degree. We also give a self-contained alternative proof of the known $Θ(N^{3/2})$ 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.
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The paper studies bounded-error quantum query complexity of tournament solutions when an oracle reveals the orientation of a queried edge. It asks how quantum access differentiates winner sets whose classical query complexities are generally $\Theta(N^2)$, and how computing a whole set compares with finding one member.
The top cycle can be computed using $O(N\log^2 N)$ queries, with an $\Omega(N)$ lower bound; this also gives the same upper bound for deciding strong connectivity of tournaments. Finding a Condorcet winner, or reporting none exists, takes $\tilde O(\sqrt N)$ queries and requires $\Omega(\sqrt N)$. Finding a strong king has complexity $\Theta(N^{3/2})$. Computing the entire Copeland set requires $\Theta(N^2)$ queries, whereas finding one maximum-out-degree vertex takes $\Theta(N^{3/2})$. Computing the uncovered set—all kings—requires $\Omega(N^{3/2})$ queries, contrasting with the known $\tilde O(\sqrt N)$ bound for finding one king.
The top-cycle algorithm constructs a Hamiltonian path classically, then uses quantum search and divide-and-conquer to locate its smallest dominant prefix through back-edge interval coverage. Lower bounds use counting promises and tournament gadgets, analyzed directly with Ambainis’s adversary method. These constructions also provide an alternative proof of the previously known maximum-degree bounds. The Condorcet algorithm combines an existing king-finding algorithm with Grover verification.
The results concern oracle-query costs rather than implementation or total runtime. The uncovered-set complexity remains between $\Omega(N^{3/2})$ and $O(N^2)$, and the top-cycle and Condorcet bounds retain polylogarithmic gaps.
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A tournament solution records a set of winners from a tournament, that is, from the outcomes of all pairwise comparisons among N 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) 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) queries, and a Condorcet winner can be found with Θ~(N) queries. Finding a strong king requires 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) 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.
The paper studies bounded-error quantum query complexity of tournament solutions when an oracle reveals the orientation of a queried edge. It asks how quantum access differentiates winner sets whose classical query complexities are generally Θ(N2), and how computing a whole set compares with finding one member.
The top cycle can be computed using O(Nlog2N) queries, with an Ω(N) lower bound; this also gives the same upper bound for deciding strong connectivity of tournaments. Finding a Condorcet winner, or reporting none exists, takes O~(N) queries and requires Ω(N). Finding a strong king has complexity Θ(N3/2). Computing the entire Copeland set requires Θ(N2) queries, whereas finding one maximum-out-degree vertex takes Θ(N3/2). Computing the uncovered set—all kings—requires Ω(N3/2) queries, contrasting with the known O~(N) bound for finding one king.
The top-cycle algorithm constructs a Hamiltonian path classically, then uses quantum search and divide-and-conquer to locate its smallest dominant prefix through back-edge interval coverage. Lower bounds use counting promises and tournament gadgets, analyzed directly with Ambainis’s adversary method. These constructions also provide an alternative proof of the previously known maximum-degree bounds. The Condorcet algorithm combines an existing king-finding algorithm with Grover verification.
The results concern oracle-query costs rather than implementation or total runtime. The uncovered-set complexity remains between Ω(N3/2) and O(N2), and the top-cycle and Condorcet bounds retain polylogarithmic gaps.
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.
A tournament is a complete directed graph. It is well known that every tournament contains at least one vertex v such that every other vertex is reachable from v by a path of length at most 2. All such vertices v are called kings of the underlying tournament. Despite active recent research in the area, the best-known upper and lower bounds on the deterministic query complexity (with query access to directions of edges) of finding a king in a tournament on n vertices are from over 20 years ago, and the bounds do not match: the best-known lower bound is Ω(n^{4/3}) and the best-known upper bound is O(n^{3/2}) [Shen, Sheng, Wu, SICOMP'03]. Our contribution is to show tight bounds (up to logarithmic factors) of Θ̃(n) and Θ̃(√n) in the randomized and quantum query models, respectively. We also study the randomized and quantum query complexities of finding a maximum out-degree vertex in a tournament.
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 O(n3/2−αk), where αk∈Θ(c−k) and c=3+17/2≈1.33, beating the previous best upper bound 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) query algorithm.
We study the minimum weight cycle problem (MWC) and the minimum weight triangle problem (MWT) in the quantum query model, for n-vertex graphs with integer edge weights in [M] given by their adjacency matrix. Roditty and Vassilevska Williams showed that the two problems are classically equivalent. We show that an exact minimum weight cycle of an undirected graph can be found with O~(n19/12) queries, while Ω(n3/2) queries are necessary. The algorithm learns only the n7/6 lightest vertex pairs, uses quantum counting to locate weight thresholds above them, applies a strict form of the Moore bound to bound the optimum and the size of each weight class, and searches the classes for the few heavy edges of an optimal cycle with nested quantum walks. A simpler version gives O~(n5/3) queries, and searching for fewer heavy edges trades queries for approximation, with a 2-approximation in O~(n8/5) queries and a tight O(logn)-approximation in O~(n3/2) queries. In contrast, MWT costs O(n9/7logM) queries, so the classical equivalence becomes a polynomial separation. For cycles of constant length we give explicit exponents, such as O~(n125/83) for the minimum weight 5-cycle, and we prove that deciding whether a graph has a k-cycle of total weight exactly zero requires Ω(n2−2/(⌊k/2⌋+1)) queries for every constant k≥4. Finally, non-adaptive learning graphs cannot beat n2 for cycles of about logn edges, and we conjecture that exact MWC on directed graphs requires n2−o(1) queries.
In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf.~Comput.'14], one's goal is to learn an unknown bit-string x∈{−1,1}n. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost O(nlogn) and a near-matching lower bound of Ω(n). Belovs [Comput.~Comp.'15] subsequently showed a tight O(n) upper bound.
We consider a natural generalization of this problem, parametrized by a subset Q⊆2[n], where an algorithm may test whether xS=b for an arbitrary S∈Q and b∈{−1,1}S of its choice, at unit cost. We show near-tight bounds when Q is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set.
All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning x is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions f:{−1,1}n→R the ratio of the maximum value of f to the maximum (over T∈Q) standard deviation of f on a subcube whose free variables are exactly T.'
To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.
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.