We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP'11].We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP'11].

Leibniz international proceedings in informaticsJan 1, 20171 citation

Derandomization of quantum algorithm for triangle finding

Guanzhong Li, Lvzhou Li

arXiv:2309.13268Information and ComputationMar 10, 20252 citations

Quantum Algorithms for Triangle Cut Sparsification

Shan Jiang, Pan Peng

Triangles capture higher-order structures in graphs and are fundamental to applications such as clustering and network analysis. To enable efficient use of such structures at scale, we study the problem of triangle cut sparsification, which aims to reduce the graph size while approximately preserving triangle counts across every cut. We investigate quantum algorithms for this problem, using triangle listing as our main technical ingredient. In particular, we present a quantum algorithm for triangle listing that, for a graph with $n$ vertices, $m$ edges, and $t$ triangles, runs in time $T_{\mathrm{q\text{-}list}} =$ $\widetilde{O}\bigl(\min(n^{5/4}t^{7/12} + n^{7/6}t^{7/9}, m + m^{3/4}t^{1/2},$ $n^{3/2}t^{1/2})\bigr)$ , improving upon the best known classical bounds over a broad range of parameters. Our algorithm is based on a heavy-light vertex partition and an extension of triangle detection via quantum walks and Grover search. Leveraging this result, we design a quantum algorithm for constructing $\varepsilon$ -triangle cut sparsifiers of size $\widetilde{O}(n/\varepsilon^2)$ in time $\widetilde{O}(T_{\mathrm{q\text{-}list}} + \sqrt{mn}/\varepsilon)$ . Finally, we demonstrate applications to clustering algorithms based on triangle-related measures and prove a lower bound of $Ω(n/\varepsilon^2)$ on the size of any $\varepsilon$ -triangle cut sparsifiers.Triangles capture higher-order structures in graphs and are fundamental to applications such as clustering and network analysis. To enable efficient use of such structures at scale, we study the problem of triangle cut sparsification, which aims to reduce the graph size while approximately preserving triangle counts across every cut. We investigate quantum algorithms for this problem, using triangle listing as our main technical ingredient. In particular, we present a quantum algorithm for triangle listing that, for a graph with $n$ vertices, $m$ edges, and $t$ triangles, runs in time $T_{\mathrm{q\text{-}list}} =$ $\widetilde{O}\bigl(\min(n^{5/4}t^{7/12} + n^{7/6}t^{7/9}, m + m^{3/4}t^{1/2},$ $n^{3/2}t^{1/2})\bigr)$, improving upon the best known classical bounds over a broad range of parameters. Our algorithm is based on a heavy-light vertex partition and an extension of triangle detection via quantum walks and Grover search. Leveraging this result, we design a quantum algorithm for constructing $\varepsilon$-triangle cut sparsifiers of size $\widetilde{O}(n/\varepsilon^2)$ in time $\widetilde{O}(T_{\mathrm{q\text{-}list}} + \sqrt{mn}/\varepsilon)$. Finally, we demonstrate applications to clustering algorithms based on triangle-related measures and prove a lower bound of $Ω(n/\varepsilon^2)$ on the size of any $\varepsilon$-triangle cut sparsifiers.

quant-pharXiv:2606.06287Jun 5, 20260 citations

n

vertices,

m

edges, and

t

triangles, runs in time

T_{\mathrm{q\text{-}list}} =

\widetilde{O}\bigl(\min(n^{5/4}t^{7/12} + n^{7/6}t^{7/9}, m + m^{3/4}t^{1/2},

n^{3/2}t^{1/2})\bigr)

, improving upon the best known classical bounds over a broad range of parameters. Our algorithm is based on a heavy-light vertex partition and an extension of triangle detection via quantum walks and Grover search. Leveraging this result, we design a quantum algorithm for constructing

\varepsilon

-triangle cut sparsifiers of size

\widetilde{O}(n/\varepsilon^2)

in time

\widetilde{O}(T_{\mathrm{q\text{-}list}} + \sqrt{mn}/\varepsilon)

. Finally, we demonstrate applications to clustering algorithms based on triangle-related measures and prove a lower bound of

Ω(n/\varepsilon^2)

on the size of any

\varepsilon

-triangle cut sparsifiers.