Quantum Algorithms for the Triangle Problem
release_e6acwugt4zeebhhbq2uenxfur4
by
Frederic Magniez, Miklos Santha, Mario Szegedy
2003
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
Õ(n^10/7) queries. The second algorithm uses Õ(n^13/10)
queries, and it is based on a design concept of Ambainis amb04 that
incorporates the benefits of quantum walks into Grover search gro96. The
first algorithm uses only O( n) qubits in its quantum subroutines,
whereas the second one uses O(n) qubits. The Triangle Problem was first treated
in bdhhmsw01, where an algorithm with O(n+√(nm)) query complexity
was presented, where m is the number of edges of G.
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