15 research outputs found
Upper bounds on quantum query complexity inspired by the Elitzur-Vaidman bomb tester
Inspired by the Elitzur-Vaidman bomb testing problem [arXiv:hep-th/9305002],
we introduce a new query complexity model, which we call bomb query complexity
. We investigate its relationship with the usual quantum query complexity
, and show that .
This result gives a new method to upper bound the quantum query complexity:
we give a method of finding bomb query algorithms from classical algorithms,
which then provide nonconstructive upper bounds on .
We subsequently were able to give explicit quantum algorithms matching our
upper bound method. We apply this method on the single-source shortest paths
problem on unweighted graphs, obtaining an algorithm with quantum
query complexity, improving the best known algorithm of [arXiv:quant-ph/0606127]. Applying this method to the maximum bipartite
matching problem gives an algorithm, improving the best known
trivial upper bound.Comment: 32 pages. Minor revisions and corrections. Regev and Schiff's proof
that P(OR) = \Omega(N) remove
Quantum Algorithms for the Most Frequently String Search, Intersection of Two String Sequences and Sorting of Strings Problems
We study algorithms for solving three problems on strings. The first one is
the Most Frequently String Search Problem. The problem is the following. Assume
that we have a sequence of strings of length . The problem is finding
the string that occurs in the sequence most often. We propose a quantum
algorithm that has a query complexity . This algorithm
shows speed-up comparing with the deterministic algorithm that requires
queries. The second one is searching intersection of two sequences
of strings. All strings have the same length . The size of the first set is
and the size of the second set is . We propose a quantum algorithm that
has a query complexity . This algorithm shows
speed-up comparing with the deterministic algorithm that requires
queries. The third problem is sorting of strings of length
. On the one hand, it is known that quantum algorithms cannot sort objects
asymptotically faster than classical ones. On the other hand, we focus on
sorting strings that are not arbitrary objects. We propose a quantum algorithm
that has a query complexity . This algorithm shows
speed-up comparing with the deterministic algorithm (radix sort) that requires
queries, where is a size of the alphabet.Comment: THe paper was presented on TPNC 201
The Dual Polynomial of Bipartite Perfect Matching
We obtain a description of the Boolean dual function of the Bipartite Perfect
Matching decision problem, as a multilinear polynomial over the Reals. We show
that in this polynomial, both the number of monomials and the magnitude of
their coefficients are at most exponential in . As an
application, we obtain a new upper bound of on the approximate degree of the bipartite perfect matching function,
improving the previous best known bound of . We deduce
that, beyond a factor, the polynomial method
cannot be used to improve the lower bound on the bounded-error quantum query
complexity of bipartite perfect matching
A Survey of Quantum Learning Theory
This paper surveys quantum learning theory: the theoretical aspects of
machine learning using quantum computers. We describe the main results known
for three models of learning: exact learning from membership queries, and
Probably Approximately Correct (PAC) and agnostic learning from classical or
quantum examples.Comment: 26 pages LaTeX. v2: many small changes to improve the presentation.
This version will appear as Complexity Theory Column in SIGACT News in June
2017. v3: fixed a small ambiguity in the definition of gamma(C) and updated a
referenc
Quantum algorithms:an overview
Quantum computers are designed to outperform standard computers by running
quantum algorithms. Areas in which quantum algorithms can be applied include
cryptography, search and optimisation, simulation of quantum systems, and
solving large systems of linear equations. Here we briefly survey some known
quantum algorithms, with an emphasis on a broad overview of their applications
rather than their technical details. We include a discussion of recent
developments and near-term applications of quantum algorithms.Comment: 17 pages; short survey to appear in npj Quantum Information. v2:
minor corrections and clarification