Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem

In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups, finding hidden subgroups of groups with small commutator subgroup and of groups admitting an elementary Abelian normal 2-subgroup of small index or with cyclic factor group.Comment: 10 page

Quantum Algorithms for the Triangle Problem

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 $G$.Comment: Several typos are fixed, and full proofs are included. Full version of the paper accepted to SODA'0

On the hitting times of quantum versus random walks

In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Further, we prove that for any reversible ergodic Markov chain $P$, the quantum hitting time of the quantum analogue of $P$ has the same order as the square root of the classical hitting time of $P$. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. Finally, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem for the 2D grid. Extending his result, we show that for any state-transitive Markov chain with unique marked state, the quantum hitting time is of the same order for both the detection and finding problems