1,314 research outputs found
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
The Quantum Query Complexity of Algebraic Properties
We present quantum query complexity bounds for testing algebraic properties.
For a set S and a binary operation on S, we consider the decision problem
whether is a semigroup or has an identity element. If S is a monoid, we
want to decide whether S is a group.
We present quantum algorithms for these problems that improve the best known
classical complexity bounds. In particular, we give the first application of
the new quantum random walk technique by Magniez, Nayak, Roland, and Santha
that improves the previous bounds by Ambainis and Szegedy. We also present
several lower bounds for testing algebraic properties.Comment: 13 pages, 0 figure
Lower Bounds on Quantum Query Complexity
Shor's and Grover's famous quantum algorithms for factoring and searching
show that quantum computers can solve certain computational problems
significantly faster than any classical computer. We discuss here what quantum
computers_cannot_ do, and specifically how to prove limits on their
computational power. We cover the main known techniques for proving lower
bounds, and exemplify and compare the methods.Comment: survey, 23 page
Quantum and Classical Tradeoffs
We propose an approach for quantifying a quantum circuit's quantumness as a
means to understand the nature of quantum algorithmic speedups. Since quantum
gates that do not preserve the computational basis are necessary for achieving
quantum speedups, it appears natural to define the quantumness of a quantum
circuit using the number of such gates. Intuitively, a reduction in the
quantumness requires an increase in the amount of classical computation, hence
giving a ``quantum and classical tradeoff''.
In this paper we present two results on this direction. The first gives an
asymptotic answer to the question: ``what is the minimum number of
non-basis-preserving gates required to generate a good approximation to a given
state''. This question is the quantum analogy of the following classical
question, ``how many fair coins are needed to generate a given probability
distribution'', which was studied and resolved by Knuth and Yao in 1976. Our
second result shows that any quantum algorithm that solves Grover's Problem of
size n using k queries and l levels of non-basis-preserving gates must have
k*l=\Omega(n)
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