We show that the quantum query complexity of detecting if an n-vertex graph contains a triangle is O(n9/7). This improves the previous best algorithm of Belovs [2] making O(n35/27) queries. For the problem of determining if an operation o : S × S → S is associative, we give an algorithm making O(|S|10/7) queries, the first improvement to the trivial O(|S|3/2) application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work [9], the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.
We show that the quantum query complexity of detecting if an n-vertex graph contains a triangle is O(n9/7). This improves the previous best algorithm of Belovs [2] making O(n35/27) queries. For the problem of determining if an operation o : S × S → S is associative, we give an algorithm making O(|S|10/7) queries, the first improvement to the trivial O(|S|3/2) application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work [9], the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.
We show that the quantum query complexity of detecting if an n-vertex graph contains a triangle is O(n9/7). This improves the previous best algorithm of Belovs making O(n35/27) queries. For the problem of determining if an operation ∘:S×S→S is associative, we give an algorithm making O(∣S∣10/7) queries, the first improvement to the trivial O(∣S∣3/2) application of Grover search. Our algorithms are designed using the learning graph framework of Belovs. We give a family of algorithms for detecting constant-sized subgraphs, which can possibly be directed and colored. These algorithms are designed in a simple high-level language; our main theorem shows how this high-level language can be compiled as a learning graph and gives the resulting complexity. The key idea to our improvements is to allow more freedom in the parameters of the database kept by the algorithm. As in our previous work, the edge slots maintained in the database are specified by a graph whose edges are the union of regular bipartite graphs, the overall structure of which mimics that of the graph of the certificate. By allowing these bipartite graphs to be unbalanced and of variable degree we obtain better algorithms.
In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity Õ(n5/4), where n denotes the number of vertices in the graph. This improves the previous upper bound O(n9/7) = O(n1.285) recently obtained by Lee, Magniez and Santha. Our result shows, for the first time, that in the quantum query complexity setting unweighted triangle finding is easier than its edge-weighted version, since for finding an edge-weighted triangle Belovs and Rosmanis proved that any quantum algorithm requires O(n9/7/ √log n) queries. Our result also illustrates some limitations of the non-adaptive learning graph approach used to obtain the previous O(n9/7) upper bound since, even over unweighted graphs, any quantum algorithm for triangle finding obtained using this approach requires v(n9/7/ √log n) queries as well. To bypass the obstacles characterized by these lower bounds, our quantum algorithm uses combinatorial ideas exploiting the graph-theoretic properties of triangle finding, which cannot be used when considering edge-weighted graphs or the non-adaptive learning graph approach.
Let H be a fixed k-vertex graph with m edges and minimum degree d>0. We use the learning graph framework of Belovs to show that the bounded-error quantum query complexity of determining if an n-vertex graph contains H as a subgraph is O(n2−2/k−t), where $ t = \max{\frac{k^2- 2(m+1)}{k(k+1)(m+1)}, \frac{2k - d - 3}{k(d+1)(m-d+2)}}$. The previous best algorithm of Magniez et al. had complexity O(n2−2/k).
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.
We develop a new framework that extends the quantum walk framework of Magniez, Nayak, Roland, and Santha, by utilizing the idea of quantum data structures to construct an efficient method of nesting quantum walks. Surprisingly, only classical data structures were considered before for searching via quantum walks. The recently proposed learning graph framework of Belovs has yielded improved upper bounds for several problems, including triangle finding and more general subgraph detection. We exhibit the power of our framework by giving a simple explicit constructions that reproduce both the O(n35/27) and O(n9/7) learning graph upper bounds (up to logarithmic factors) for triangle finding, and discuss how other known upper bounds in the original learning graph framework can be converted to algorithms in our framework. We hope that the ease of use of this framework will lead to the discovery of new upper bounds.