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Register a new userGeneralization of Grovers Algorithm to Multiobject Search in Quantum Computing, Part I: Continuous Time and Discrete Time
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L. K. Grovers search algorithm in quantum computing gives an optimal, quadratic speedup in the search for a single object in a large unsorted database. In this paper, we generalize Grovers algorithm in a Hilbert-space framework for both continuous and discrete time cases that isolates its geometrical essence to the case where more than one object satisfies the search criterion.
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There are major advantages in a newer version of Grovers quantum algorithm utilizing a general unitary transformation in the search of a single object in a large unsorted database. In this paper, we generalize this algorithm to multiobject search. We show the techniques to achieve the reduction of the problem to one on an invariant subspace of dimension just equal to two.
We question whether the measurement based quantum computing algorithm is in fact Grovers algorithm or simply a similar oracular search method. The two algorithms share several qualitative features especially in the case of the trivial 4 element search, which is the largest size photonic search algorithm that has been experimentally implemented to date. This has led some to refer to both substantiations as Grovers algorithm. We compare multiple features of the two algorithms including the behavior of the oracle tags and the entanglement dynamics, both qualitatively and quantitatively. We find significant and fundamental differences in the operation of the two algorithms, particularly in cases involving searches on more than four elements.
We find that the Measurement Based Quantum Computing (MBQC) search algorithm on an unsorted list is not the same as Grovers search algorithm (GSA).
Grovers quantum algorithm improves any classical search algorithm. We show how random Gaussian noise at each step of the algorithm can be modelled easily because of the exact recursion formulas available for computing the quantum amplitude in Grovers algorithm. We study the algorithms intrinsic robustness when no quantum correction codes are used, and evaluate how much noise the algorithm can bear with, in terms of the size of the phone book and a desired probability of finding the correct result. The algorithm loses efficiency when noise is added, but does not slow down. We also study the maximal noise under which the iterated quantum algorithm is just as slow as the classical algorithm. In all cases, the width of the allowed noise scales with the size of the phone book as N^-2/3.
We present new results on an optical implementation of Grovers quantum search algorithm. This extends previous work in which the transverse spatial mode of a light beam oscillates between a broad initial input shape and a highly localized spike, which reveals the position of the tagged item. The spike reaches its maximum intensity after $simsqrt N$ round trips in a cavity equipped with two phase plates, where $N$ is the ratio of the surface area of the original beam and the area of the phase spot or tagged item. In our redesigned experiment the search space is now two-dimensional. In the time domain we demonstrate for the first time a multiple item search where the items appear directly as bright spots on the images of a gated camera. In a complementary experiment we investigate the searching cavity in the frequency domain. The oscillatory nature of the search algorithm can be seen as a splitting of cavity eigenmodes, each of which concentrates up to 50% of its power in the bright spot corresponding to the solution.
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