16,441 research outputs found
Group Theoretical Formulation of Quantum Partial Search Algorithm
Searching and sorting used as a subroutine in many important algorithms.
Quantum algorithm can find a target item in a database faster than any
classical algorithm. One can trade accuracy for speed and find a part of the
database (a block) containing the target item even faster, this is partial
search. An example is the following: exact address of the target item is given
by a sequence of many bits, but we need to know only some of them. More
generally partial search considers the following problem: a database is
separated into several blocks. We want to find a block with the target item,
not the target item itself. In this paper we reformulate quantum partial search
algorithm in terms of group theory.Comment: 12 page
Simple Algorithm for Partial Quantum Search
Quite often in database search, we only need to extract portion of the
information about the satisfying item. Recently Radhakrishnan & Grover [RG]
considered this problem in the following form: the database of items was
divided into equally sized blocks. The algorithm has just to find the block
containing the item of interest. The queries are exactly the same as in the
standard database search problem. [RG] invented a quantum algorithm for this
problem of partial search that took about fewer iterations
than the quantum search algorithm. They also proved that the best any quantum
algorithm could do would be to save iterations. The main
limitation of the algorithm was that it involved complicated analysis as a
result of which it has been inaccessible to most of the community. This paper
gives a simple analysis of the algorithm. This analysis is based on three
elementary observations about quantum search, does not require a single
equation and takes less than 2 pages.Comment: 3 pages, 3 figure
Optimization of Partial Search
Quantum Grover search algorithm can find a target item in a database faster
than any classical algorithm. One can trade accuracy for speed and find a part
of the database (a block) containing the target item even faster, this is
partial search. A partial search algorithm was recently suggested by Grover and
Radhakrishnan. Here we optimize it. Efficiency of the search algorithm is
measured by number of queries to the oracle. The author suggests new version of
Grover-Radhakrishnan algorithm which uses minimal number of queries to the
oracle. The algorithm can run on the same hardware which is used for the usual
Grover algorithm.Comment: 5 page
Quantum Multi-object Search Algorithm with the Availability of Partial Information
Consider the unstructured search of an unknown number l of items in a large
unsorted database of size N. The multi-object quantum search algorithm consists
of two parts. The first part of the algorithm is to generalize Grover's
single-object search algorithm to the multi-object case and the second part is
to solve a counting problem to determine l.
In this paper, we study the multi-object quantum search algorithm (in
continuous time), but in a more structured way by taking into account the
availability of partial information. The modeling of available partial
information is done simply by the combination of several prescribed, possibly
overlapping, information sets with varying weights to signify the reliability
of each set. The associated statistics is estimated and the algorithm
efficiency and complexity are analyzed.
Our analysis shows that the search algorithm described here may not be more
efficient than the unstructured (generalized) multi-object Grover search if
there is ``misplaced confidence''. However, if the information sets have a
``basic confidence'' property in the sense that each information set contains
at least one search item, then a quadratic speedup holds on a much smaller data
space, which further expedite the quantum search for the first item.Comment: 17 pages, 1 figur
Optimal Database Search: Waves and Catalysis
Grover's database search algorithm, although discovered in the context of
quantum computation, can be implemented using any system that allows
superposition of states. A physical realization of this algorithm is described
using coupled simple harmonic oscillators, which can be exactly solved in both
classical and quantum domains. Classical wave algorithms are far more stable
against decoherence compared to their quantum counterparts. In addition to
providing convenient demonstration models, they may have a role in practical
situations, such as catalysis.Comment: 4 pages (v2) 6 pages, RevTeX4. Title changed. Substantially expanded
to include stability considerations, quantum domain analysis, application to
catalysis. (v3) Version accepted for publication. (v4) Error in Eqs.(10,11)
corrected by replacing \omega by \omega^2. No change in implementation and
applicatio
Generalized Quantum Search with Parallelism
We generalize Grover's unstructured quantum search algorithm to enable it to
use an arbitrary starting superposition and an arbitrary unitary matrix
simultaneously. We derive an exact formula for the probability of the
generalized Grover's algorithm succeeding after n iterations. We show that the
fully generalized formula reduces to the special cases considered by previous
authors. We then use the generalized formula to determine the optimal strategy
for using the unstructured quantum search algorithm. On average the optimal
strategy is about 12% better than the naive use of Grover's algorithm. The
speedup obtained is not dramatic but it illustrates that a hybrid use of
quantum computing and classical computing techniques can yield a performance
that is better than either alone. We extend the analysis to the case of a
society of k quantum searches acting in parallel. We derive an analytic formula
that connects the degree of parallelism with the optimal strategy for
k-parallel quantum search. We then derive the formula for the expected speed of
k-parallel quantum search.Comment: 14 pages, 2 figure
Qdensity - a Mathematica Quantum Computer Simulation
This Mathematica 5.2 package~\footnote{QDENSITY is available at
http://www.pitt.edu/~tabakin/QDENSITY} is a simulation of a Quantum Computer.
The program provides a modular, instructive approach for generating the basic
elements that make up a quantum circuit. The main emphasis is on using the
density matrix, although an approach using state vectors is also implemented in
the package. The package commands are defined in {\it Qdensity.m} which
contains the tools needed in quantum circuits, e.g. multiqubit kets,
projectors, gates, etc. Selected examples of the basic commands are presented
here and a tutorial notebook, {\it Tutorial.nb} is provided with the package
(available on our website) that serves as a full guide to the package. Finally,
application is made to a variety of relevant cases, including Teleportation,
Quantum Fourier transform, Grover's search and Shor's algorithm, in separate
notebooks: {\it QFT.nb}, {\it Teleportation.nb}, {\it Grover.nb} and {\it
Shor.nb} where each algorithm is explained in detail. Finally, two examples of
the construction and manipulation of cluster states, which are part of ``one
way computing" ideas, are included as an additional tool in the notebook {\it
Cluster.nb}. A Mathematica palette containing most commands in QDENSITY is also
included: {\it QDENSpalette.nb} .Comment: The Mathematica 5+ package is available at:
http://www.pitt.edu/~tabakin/QDENSITY/QDENSITY.htm Minor corrections,
accepted in Computer Physics Communication
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