Abstract L. K. Grover's search algorithm in quantum computing gives an optimal, square-root speedupin the search for a single object in a large unsorted database. In this paper, we expound Grover's algorithm in a Hilbert-space framework that isolates its geometrical essence, and we generalizeit to the case where more than one object satisfies the search criterion.

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 Deutsch-Jozsa, 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

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.

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 non-abelian hidden subgroup problem (HSP). But we then go on to show that the standard non-abelian 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 non-abelian QHS algorithm which is equivalent to Grover’s algorithm. Contents

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

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

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.

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 ([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 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.

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 discrete-time operations(e.g., Walsh-Hadamard). 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 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