Frédéric Magniez

This paper considers the triangle finding problem in the CONGEST model of distributed computing. Recent works by Izumi and Le Gall (PODC’17), Chang, Pettie and Zhang (SODA’19) and Chang and Saranurak (PODC’19) have successively reduced the classical round complexity of triangle finding (as well as triangle listing) from the trivial upper bound $O(n)$ to $\tilde O(n^{1/3})$, where~$n$ denotes the number of vertices in the graph. In this paper we present a quantum distributed algorithm that solves the triangle finding problem in $\tilde O(n^{1/4})$ rounds in the CONGEST model. This gives another example of quantum algorithm beating the best known classical algorithms in distributed computing. Our result also exhibits an interesting phenomenon: while in the classical setting the best known upper bounds for the triangle finding and listing problems are identical, in the quantum setting the round complexities of these two problems are now $\tilde O(n^{1/4})$ and $\tilde \Theta(n^{1/3})$, respectively. Our result thus shows that triangle finding is easier than triangle listing in the quantum CONGEST model.

We present two new quantum algorithms that either finds a triangle in an undirected graph G on n nodes, or rejects if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes \tilde{O}(n^{10/7}) queries. The second algorithm uses O(n^{13/10}) queries, and it is based on a new design concept of Ambainis that incorporates the benefits of quantum walks into Grover search. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. Note: joint work with Miklos Santha and Mario Szegedy