Quantum query complexity of triangle finding

Close the gap between the Ω(n) lower bound and O(n^1.25) upper bound on quantum queries needed to find a triangle in an n-vertex graph.

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1	# Quantum query complexity of triangle finding
2
3	How many questions of the form "is $\{u,v\}$ an edge?" must a quantum algorithm ask to find a triangle in an $n$-vertex graph? Classical algorithms need about $n^2$ such queries, and the best quantum algorithm uses $O(n^{5/4})$, but the best lower bound is only $\Omega(n)$. Triangle finding has driven the development of quantum walks and learning graphs, and closing this gap is a basic open problem in quantum query complexity.
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5	## Problem
6
7	Let $G$ be a simple graph on the vertex set $\{1,\dots,n\}$ with adjacency matrix $A$. A quantum algorithm may access $A$ only through the oracle
8
9	$$O_A\lvert u,v,z\rangle=\lvert u,v,z\oplus A_{uv}\rangle ,$$
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11	which it can query in superposition. It must output three vertices $u,v,w$ with $A_{uv}=A_{vw}=A_{wu}=1$ if a triangle exists, and report that $G$ is triangle-free otherwise, with success probability at least $2/3$ on every graph. Let $Q_\triangle(n)$ be the least number of queries such an algorithm needs on $n$-vertex graphs. It is known that
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13	$$\Omega(n)\le Q_\triangle(n)\le O(n^{5/4}).$$
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15	Question. Determine the growth of $Q_\triangle(n)$. In particular, is there an algorithm with $O(n^{5/4-\varepsilon})$ queries, or a lower bound $\Omega(n^{1+\varepsilon})$, for some constant $\varepsilon>0$?
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17	## Background
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19	The quantum query complexity of triangle finding was first raised by Buhrman, Dürr, Heiligman, Høyer, Magniez, Santha and de Wolf, whose algorithm uses $O(n+\sqrt{nm})$ queries on graphs with $m$ edges [1, 2]. Grover search alone gives $O(n^{3/2})$ queries, while classical randomized algorithms need $\Theta(n^2)$ [4]. The upper bound then fell with each new technique: $\tilde O(n^{13/10})$ with quantum walks (Magniez, Santha and Szegedy), $O(n^{35/27})$ with learning graphs (Belovs), and $O(n^{9/7})$ (Lee, Magniez and Santha) [2, 4]. The problem is also closely tied to matrix multiplication [2].
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21	## What is known
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23	- Upper bound $O(n^{5/4})$: Le Gall's $\tilde O(n^{5/4})$-query algorithm (2014) exploits combinatorial properties of unweighted graphs [4], and Carette, Laurière and Magniez removed the logarithmic factors with extended learning graphs (2017) [5].
24	- Lower bound $\Omega(n)$: this trivial bound is the best known [4]. Magniez, Santha and Szegedy observed that the adversary methods available in 2005 cannot prove more for this problem, because a triangle is a certificate of constant size [2].
25	- For an edge-weighted version, the triangle-sum problem, Belovs and Rosmanis proved a lower bound of $\Omega(n^{9/7}/\sqrt{\log n})$ queries [3]. Non-adaptive learning graphs cannot tell the two versions apart, so for unweighted graphs they cannot beat $n^{9/7}$ by more than a $\sqrt{\log n}$ factor; the $n^{5/4}$ algorithms get around this by using the structure of unweighted graphs, showing that unweighted triangle finding is the easier problem [3, 4].
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27	## What would count as progress
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29	- An algorithm with $O(n^{5/4-\varepsilon})$ queries for a constant $\varepsilon>0$. Algorithms in the learning-graph frameworks of [3, 5] come with explicit complexity formulas that can be checked directly.
30	- A lower bound $\Omega(n^{1+\varepsilon})$ for unweighted triangle finding. The general adversary bound characterizes quantum query complexity [4], so such a bound can in principle be certified by an explicit adversary matrix; it must go beyond the methods discussed in [2].
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32	## References
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34	1. H. Buhrman, C. Dürr, M. Heiligman, P. Høyer, F. Magniez, M. Santha and R. de Wolf, "Quantum algorithms for element distinctness", SIAM Journal on Computing 34(6), 1324–1330 (2005). [doi:10.1137/S0097539702402780](https://doi.org/10.1137/S0097539702402780)
35	2. F. Magniez, M. Santha and M. Szegedy, "Quantum algorithms for the triangle problem", SIAM Journal on Computing 37(2), 413–424 (2007). [doi:10.1137/050643684](https://doi.org/10.1137/050643684)
36	3. A. Belovs and A. Rosmanis, "On the power of non-adaptive learning graphs", Computational Complexity 23, 323–354 (2014). [doi:10.1007/s00037-014-0084-1](https://doi.org/10.1007/s00037-014-0084-1)
37	4. F. Le Gall, "Improved quantum algorithm for triangle finding via combinatorial arguments", in 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), 216–225 (2014). [doi:10.1109/FOCS.2014.31](https://doi.org/10.1109/FOCS.2014.31)
38	5. T. Carette, M. Laurière and F. Magniez, "Extended learning graphs for triangle finding", in 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), LIPIcs 66, 20 (2017). [doi:10.4230/LIPIcs.STACS.2017.20](https://doi.org/10.4230/LIPIcs.STACS.2017.20)
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40	## Source
41
42	Imported from [QIQCOP Zoo op_0c17d9fef967858d](https://qiqc-op.com/problem/op_0c17d9fef967858d/), curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.