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Grover's Algorithm for Multiobject Search in Quantum Computing
"... Abstract L. K. Grover's search algorithm in quantum computing gives an optimal, squareroot speedupin the search for a single object in a large unsorted database. In this paper, we expound Grover's algorithm in a Hilbertspace framework that isolates its geometrical essence, and we generalizeit to t ..."
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Abstract L. K. Grover's search algorithm in quantum computing gives an optimal, squareroot speedupin the search for a single object in a large unsorted database. In this paper, we expound Grover's algorithm in a Hilbertspace framework that isolates its geometrical essence, and we generalizeit to the case where more than one object satisfies the search criterion.
Quantum hidden subgroup algorithms on free groups, (in preparation
"... Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In thi ..."
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Abstract. One of the most promising and versatile approaches to creating new quantum algorithms is based on the quantum hidden subgroup (QHS) paradigm, originally suggested by Alexei Kitaev. This class of quantum algorithms encompasses the DeutschJozsa, Simon, Shor algorithms, and many more. In this paper, our strategy for finding new quantum algorithms is to decompose Shor’s quantum factoring algorithm into its basic primitives, then to generalize these primitives, and finally to show how to reassemble them into new QHS algorithms. Taking an ”alphabetic building blocks approach, ” we use these primitives to form an ”algorithmic toolkit ” for the creation of new quantum algorithms, such as wandering Shor algorithms, continuous Shor algorithms, the quantum circle algorithm, the dual Shor algorithm, a QHS algorithm for Feynman integrals, free QHS algorithms, and more. Toward the end of this paper, we show how Grover’s algorithm is most surprisingly “almost ” a QHS algorithm, and how this result suggests the possibility of an even more complete ”algorithmic tookit ” beyond the QHS algorithms. Contents
Singlestep quantum search using problem structure.” eprint quantph/9812049
"... 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 kSAT allows determining the asymptotic average behavior of these al ..."
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Cited by 4 (2 self)
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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 kSAT 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.
IS GROVER’S ALGORITHM A QUANTUM HIDDEN SUBGROUP ALGORITHM?
, 2006
"... Abstract. The arguments given in this paper suggest that Grover’s and Shor’s algorithms are more closely related than one might at first expect. Specifically, we show that Grover’s algorithm can be viewed as a quantum algorithm which solves a nonabelian hidden subgroup problem (HSP). But we then go ..."
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Abstract. The arguments given in this paper suggest that Grover’s and Shor’s algorithms are more closely related than one might at first expect. Specifically, we show that Grover’s algorithm can be viewed as a quantum algorithm which solves a nonabelian hidden subgroup problem (HSP). But we then go on to show that the standard nonabelian quantum hidden subgroup (QHS) algorithm can not find a solution to this particular HSP. This leaves open the question as to whether or not there is some modification of the standard nonabelian QHS algorithm which is equivalent to Grover’s algorithm. Contents
ConstantTime Quantum Algorithm For The Unstructured Search Problem
, 2008
"... Given an item and a list of values of size N. It is required to decide if such item exists in the list. Classical computer can search for the item in O(N). The best known quantum algorithm can do the job in O ( √ N). In this paper, a quantum algorithm will be proposed that can search an unstructure ..."
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Given an item and a list of values of size N. It is required to decide if such item exists in the list. Classical computer can search for the item in O(N). The best known quantum algorithm can do the job in O ( √ N). In this paper, a quantum algorithm will be proposed that can search an unstructured list in O(1) to get the YES/NO answer with certainty. 1
Strength and Weakness in Grover’s Quantum Search Algorithm
, 2008
"... Grover’s quantum search algorithm is considered as one of the milestone in the field of quantum computing. The algorithm can search for a single match in a database with N records in O ( √ N) assuming that the item must exist in the database with quadratic speedup over the best known classical algo ..."
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Grover’s quantum search algorithm is considered as one of the milestone in the field of quantum computing. The algorithm can search for a single match in a database with N records in O ( √ N) assuming that the item must exist in the database with quadratic speedup over the best known classical algorithm. This review paper discusses the performance of Grover’s algorithm in case of multiple matches where the problem is expected to be easier. Unfortunately, we will find that the algorithm will fail for M> 3N/4, where M is the number of matches in the list. 1
Noise in Grover’s Quantum Search Algorithm B. PabloNorman and M. RuizAltaba
, 1999
"... Grover’s 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 Grover’s algorithm. We study the algorithm’s intrinsic r ..."
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Grover’s 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 Grover’s algorithm. We study the algorithm’s intrinsic robustess 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. Typeset using REVTEX 1 I.
Quantum Multiobject Search Algorithm with the Availability of Partial Information
, 2000
"... Consider the unstructured search of an unknown number l of items in a large unsorted database of size N. The multiobject quantum search algorithm consists of two parts. The first part of the algorithm is to generalize Grover’s singleobject search algorithm to the multiobject case ([3, 4, 5, 6, 7] ..."
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Consider the unstructured search of an unknown number l of items in a large unsorted database of size N. The multiobject quantum search algorithm consists of two parts. The first part of the algorithm is to generalize Grover’s singleobject search algorithm to the multiobject case ([3, 4, 5, 6, 7]) and the second part is to solve a counting problem to determine l ([4, 14]). In this paper, we study the multiobject 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) multiobject 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.
Ion Trap Proposal for Quantum Search
, 2002
"... In this letter, we show that the laser Hamiltonian can perform the quantum search. We also show that the process of quantum search is a resonance between the initial state and the target state, which implies that Nature already has a quantum search system to use a transition of energy. In addition, ..."
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In this letter, we show that the laser Hamiltonian can perform the quantum search. We also show that the process of quantum search is a resonance between the initial state and the target state, which implies that Nature already has a quantum search system to use a transition of energy. In addition, we provide the particular scheme to implement the quantum search algorithm based on a trapped ion. Quantum computation has been in the spotlight, supplying the solution for the problems which are intractable in the context of classical physics. The quantum factorization algorithm and the quantum search algorithm are the good examples.[1] In particular, the quantum search algorithm provides the quadratic speedup in solving a search problem. Here, the search problem is to find the target of the unstructured N itmes. Grover’s fast quantum search algorithm is composed of discretetime operations(e.g., WalshHadamard). When we use Grover’s algorithm, we need O ( √ N) iterations of the Grover operator. There is the analog quantum search algorithm which is based on the Hamiltonian evolution. Farhi and Gutmann proposed the quantum search Hamiltonian.[5] Moreover, Fenner provided another
Hypothesis elimination on a quantum computer ∗
, 2004
"... Hypothesis elimination is a special case of Bayesian updating, where each piece of new data rules out a set of prior hypotheses. We describe how to use Grover’s algorithm to perform hypothesis elimination for a class of probability distributions encoded on a register of qubits, and establish a lower ..."
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Hypothesis elimination is a special case of Bayesian updating, where each piece of new data rules out a set of prior hypotheses. We describe how to use Grover’s algorithm to perform hypothesis elimination for a class of probability distributions encoded on a register of qubits, and establish a lower bound on the required computational resources. 1