4,148 research outputs found
Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision
The quantum query complexity of Boolean matrix multiplication is typically
studied as a function of the matrix dimension, n, as well as the number of 1s
in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values
of \ell. This is an improvement over previous algorithms for all values of
\ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps
n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing
that our algorithm is essentially tight.
We first reduce Boolean matrix multiplication to several instances of graph
collision. We then provide an algorithm that takes advantage of the fact that
the underlying graph in all of our instances is very dense to find all graph
collisions efficiently
Weak Parity
We study the query complexity of Weak Parity: the problem of computing the
parity of an n-bit input string, where one only has to succeed on a 1/2+eps
fraction of input strings, but must do so with high probability on those inputs
where one does succeed. It is well-known that n randomized queries and n/2
quantum queries are needed to compute parity on all inputs. But surprisingly,
we give a randomized algorithm for Weak Parity that makes only
O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only
O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of
Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove
lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in
the quantum case for any eps>0. We show that improving our lower bounds is
intimately related to two longstanding open problems about Boolean functions:
the Sensitivity Conjecture, and the relationships between query complexity and
polynomial degree.Comment: 18 page
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
An optimal quantum algorithm for the oracle identification problem
In the oracle identification problem, we are given oracle access to an
unknown N-bit string x promised to belong to a known set C of size M and our
task is to identify x. We present a quantum algorithm for the problem that is
optimal in its dependence on N and M. Our algorithm considerably simplifies and
improves the previous best algorithm due to Ambainis et al. Our algorithm also
has applications in quantum learning theory, where it improves the complexity
of exact learning with membership queries, resolving a conjecture of Hunziker
et al.
The algorithm is based on ideas from classical learning theory and a new
composition theorem for solutions of the filtered -norm semidefinite
program, which characterizes quantum query complexity. Our composition theorem
is quite general and allows us to compose quantum algorithms with
input-dependent query complexities without incurring a logarithmic overhead for
error reduction. As an application of the composition theorem, we remove all
log factors from the best known quantum algorithm for Boolean matrix
multiplication.Comment: 16 pages; v2: minor change
Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments
In this paper we present a quantum algorithm solving the triangle finding
problem in unweighted graphs with query complexity , where
denotes the number of vertices in the graph. This improves the previous
upper bound recently obtained by Lee, Magniez and
Santha. Our result shows, for the first time, that in the quantum query
complexity setting unweighted triangle finding is easier than its edge-weighted
version, since for finding an edge-weighted triangle Belovs and Rosmanis proved
that any quantum algorithm requires queries.
Our result also illustrates some limitations of the non-adaptive learning graph
approach used to obtain the previous upper bound since, even over
unweighted graphs, any quantum algorithm for triangle finding obtained using
this approach requires queries as well. To
bypass the obstacles characterized by these lower bounds, our quantum algorithm
uses combinatorial ideas exploiting the graph-theoretic properties of triangle
finding, which cannot be used when considering edge-weighted graphs or the
non-adaptive learning graph approach.Comment: 17 pages, to appear in FOCS'14; v2: minor correction
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