15 research outputs found
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
) in an undirected graph on nodes, or reject if is triangle
free. The first algorithm uses combinatorial ideas with Grover Search and makes
queries. The second algorithm uses
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 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 query complexity
was presented, where is the number of edges of .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 , the
quantum hitting time of the quantum analogue of has the same order as the
square root of the classical hitting time of . 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