Quantum Algorithms for the Triangle Problem
Abstract
We present two new quantum algorithms that either find a triangle (a copy of $K_{3}$) in an undirected graph $G$ on $n$ nodes, or reject 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 $\tilde{O}(n^{13/10})$ queries, and it is based on a design concept of Ambainis~\cite{amb04} that incorporates the benefits of quantum walks into Grover search~\cite{gro96}. The first algorithm uses only $O(\log n)$ qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in~\cite{bdhhmsw01}, where an algorithm with $O(n+\sqrt{nm})$ query complexity was presented, where $m$ is the number of edges of $G$.
 Publication:

arXiv eprints
 Pub Date:
 October 2003
 arXiv:
 arXiv:quantph/0310134
 Bibcode:
 2003quant.ph.10134M
 Keywords:

 Quantum Physics
 EPrint:
 Several typos are fixed, and full proofs are included. Full version of the paper accepted to SODA'05