4,008 research outputs found
Quantum search algorithms on a regular lattice
Quantum algorithms for searching one or more marked items on a d-dimensional
lattice provide an extension of Grover's search algorithm including a spatial
component. We demonstrate that these lattice search algorithms can be viewed in
terms of the level dynamics near an avoided crossing of a one-parameter family
of quantum random walks. We give approximations for both the level-splitting at
the avoided crossing and the effectively two-dimensional subspace of the full
Hilbert space spanning the level crossing. This makes it possible to give the
leading order behaviour for the search time and the localisation probability in
the limit of large lattice size including the leading order coefficients. For
d=2 and d=3, these coefficients are calculated explicitly. Closed form
expressions are given for higher dimensions
Realization of generalized quantum searching using nuclear magnetic resonance
According to the theoretical results, the quantum searching algorithm can be
generalized by replacing the Walsh-Hadamard(W-H) transform by almost any
quantum mechanical operation. We have implemented the generalized algorithm
using nuclear magnetic resonance techniques with a solution of chloroform
molecules. Experimental results show the good agreement between theory and
experiment.Comment: 11 pages,3 figure. Accepted by Phys. Rev. A. Scheduled Issue: 01 Mar
200
Spatial quantum search in a triangular network
The spatial search problem consists in minimizing the number of steps
required to find a given site in a network, under the restriction that only
oracle queries or translations to neighboring sites are allowed. We propose a
quantum algorithm for the spatial search problem on a triangular lattice with N
sites and torus-like boundary conditions. The proposed algortithm is a special
case of the general framework for abstract search proposed by Ambainis, Kempe
and Rivosh [AKR05] (AKR) and Tulsi [Tulsi08], applied to a triangular network.
The AKR-Tulsi formalism was employed to show that the time complexity of the
quantum search on the triangular lattice is O(sqrt(N logN)).Comment: 10 pages, 4 Postscript figures, uses sbc-template.sty, appeared in
Annals of WECIQ 2010, III Workshop of Quantum Computation and Quantum
Informatio
Grover's Quantum Search Algorithm for an Arbitrary Initial Mixed State
The Grover quantum search algorithm is generalized to deal with an arbitrary
mixed initial state. The probability to measure a marked state as a function of
time is calculated, and found to depend strongly on the specific initial state.
The form of the function, though, remains as it is in the case of initial pure
state. We study the role of the von Neumann entropy of the initial state, and
show that the entropy cannot be a measure for the usefulness of the algorithm.
We give few examples and show that for some extremely mixed initial states
carrying high entropy, the generalized Grover algorithm is considerably faster
than any classical algorithm.Comment: 4 pages. See http://www.cs.technion.ac.il/~danken/MSc-thesis.pdf for
extended discussio
A General SU(2) Formulation for Quantum Searching with Certainty
A general quantum search algorithm with arbitrary unitary transformations and
an arbitrary initial state is considered in this work. To serach a marked state
with certainty, we have derived, using an SU(2) representation: (1) the
matching condition relating the phase rotations in the algorithm, (2) a concise
formula for evaluating the required number of iterations for the search, and
(3) the final state after the search, with a phase angle in its amplitude of
unity modulus. Moreover, the optimal choices and modifications of the phase
angles in the Grover kernel is also studied.Comment: 8 pages, 2 figure
Characterization of pure quantum states of multiple qubits using the Groverian entanglement measure
The Groverian entanglement measure, G(psi), is applied to characterize a
variety of pure quantum states |psi> of multiple qubits. The Groverian measure
is calculated analytically for certain states of high symmetry, while for
arbitrary states it is evaluated using a numerical procedure. In particular, it
is calculated for the class of Greenberger-Horne-Zeilinger states, the W states
as well as for random pure states of n qubits. The entanglement generated by
Grover's algorithm is evaluated by calculating G(psi) for the intermediate
states that are obtained after t Grover iterations, for various initial states
and for different sets of the marked states.Comment: 28 pages, 5 figure
Indications of superconductivity in doped highly oriented pyrolytic graphite
We have observed possible superconductivity using standard resistance vs.
temperature techniques in phosphorous ion implanted Highly Oriented Pyrolytic
Graphite. The onset appears to be above 100 K and quenching by an applied
magnetic field has been observed. The four initial boron implanted samples
showed no signs of becoming superconductive whereas all four initial and eight
subsequent samples that were implanted with phosphorous showed at least some
sign of the existence of small amounts of the possibly superconducting phases.
The observed onset temperature is dependent on both the number of electron
donors present and the amount of damage done to the graphene sub-layers in the
Highly Oriented Pyrolytic Graphite samples. As a result the data appears to
suggest that the potential for far higher onset temperatures in un-damaged
doped graphite exists.Comment: 7 pages, 1 table, 5 figures, 11 references, Acknowledgments section
was correcte
Single-Step Quantum Search Using Problem Structure
The structure of satisfiability problems is used to improve search algorithms
for quantum computers and reduce their required coherence times by using only a
single coherent evaluation of problem properties. The structure of random k-SAT
allows determining the asymptotic average behavior of these algorithms, showing
they improve on quantum algorithms, such as amplitude amplification, that
ignore detailed problem structure but remain exponential for hard problem
instances. Compared to good classical methods, the algorithm performs better,
on average, for weakly and highly constrained problems but worse for hard
cases. The analytic techniques introduced here also apply to other quantum
algorithms, supplementing the limited evaluation possible with classical
simulations and showing how quantum computing can use ensemble properties of NP
search problems.Comment: 39 pages, 12 figures. Revision describes further improvement with
multiple steps (section 7). See also
http://www.parc.xerox.com/dynamics/www/quantum.htm
Algebraic analysis of quantum search with pure and mixed states
An algebraic analysis of Grover's quantum search algorithm is presented for
the case in which the initial state is an arbitrary pure quantum state of n
qubits. This approach reveals the geometrical structure of the quantum search
process, which turns out to be confined to a four-dimensional subspace of the
Hilbert space. This work unifies and generalizes earlier results on the time
evolution of the amplitudes during the quantum search, the optimal number of
iterations and the success probability. Furthermore, it enables a direct
generalization to the case in which the initial state is a mixed state,
providing an exact formula for the success probability.Comment: 13 page
The Precise Formula in a Sine Function Form of the norm of the Amplitude and the Necessary and Sufficient Phase Condition for Any Quantum Algorithm with Arbitrary Phase Rotations
In this paper we derived the precise formula in a sine function form of the
norm of the amplitude in the desired state, and by means of he precise formula
we presented the necessary and sufficient phase condition for any quantum
algorithm with arbitrary phase rotations. We also showed that the phase
condition: identical rotation angles, is a sufficient but not a necessary phase
condition.Comment: 16 pages. Modified some English sentences and some proofs. Removed a
table. Corrected the formula for kol on page 10. No figure
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