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In this paper, we propose a new quantum algorithm for solving the minimum searching problem. This algorithm has the same order of time and space complexities as the algorithm proposed by Dürr and Høyer, but it provides a quadratic reduction in the number of measurements. In addition to the correctness and complexity analysis of the algorithm, we present simulation results considering an NP-hard problem.

It is well-known that Shor's factorization algorithm, Simon's period-finding algorithm, and Deutsch's original XOR algorithm can all be formulated as solutions to a hidden subgroup problem. Here the salient features of the information-processing in the three algorithms are presented from a different perspective, in terms of the way in which the algorithms exploit the non-Boolean quantum logic represented by the projective geometry of Hilbert space. From this quantum logical perspective, the XOR algorithm appears directly as a special case of Simon's algorithm, and all three algorithms can be seen as exploiting the non-Boolean logic represented by the subspace structure of Hilbert space in a similar way. Essentially, a global property of a function (such as a period, or a disjunctive property) is encoded as a subspace in Hilbert space representing a quantum proposition, which can then be efficiently distinguished from alternative propositions, corresponding to alternative global properties, by a measurement (or sequence of measurements) that identifies the target proposition as the proposition represented by the subspace containing the final state produced by the algorithm.

In the field of Quantum computing, the Peter Shorgave an important algorithm known as Shor's factorization algorithm. With the existing coding it has an issue of finding order and periods finding. Search of new quantum numbers has been already on to strengthen the quantum computing. The development of quantum computer is also slow due to limited availability of simulation. In this paper, we have reported new quantum numbers using Shor's algorithm and simulation. The graphical representation of numbers are shown in this paper.

Seit der Entdeckung von effizienten Quantenalgorithmen für das Faktorisieren von ganzen Zahlen und für die Berechnung von diskreten Logarithmen durch Shor in 1994 wuchs das Interesse an Quantenalgorithmen innerhalb der theoretischen Informatik- und der Physik-Gemeinde. Überraschenderweise ist die Anzahl der Quantenalgorithmen, die bisher entdeckt wurden, vergleichsweise gering, obwohl die Anzahl der Forscher, welche an diesem Gebiet arbeiten, stark angestiegen ist. Die Aufgabe, neue Quantenalgorithmen zu entwerfen, hat sich in der Tat als extrem schwierig herausgestellt. Wir geben einen kurzen Überblick über eine Auswahl von Problemen, für welche effiziente Quantenalgorithmen bekannt sind, und beschreiben kurz die zu Grunde liegenden Ideen. Ein großes Gebiet, bei dem ein Quantenrechner allen klassischen Rechnern überlegen zu sein scheint, sind die verborgenen Untergruppenprobleme. Wir erklären deren Relevanz und geben Motivation für diese abstrakte Problemklasse.

We deal with the problem of finding a maximum of a function from the Holder class on a quantum computer. We show matching lower and upper bounds on the complexity of this problem. We prove upper bounds by constructing an algorithm that uses a pre-existing quantum algorithm for finding maximum of a discrete sequence. To prove lower bounds we use results for finding the logical OR of sequence of bits. We show that quantum computation yields a quadratic speed-up over deterministic and randomized algorithms.

We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log |G|) number of calls to the oracle. This is exponentially better than the best classical algorithm. However our quantum algorithm requires exponential time, as in the classical case. Our algorithm utilizes a new technique for constructing error-free algorithms for non-decision problems on quantum computers., To appear in Information Processing Letters (IPL)

We obtain the strongest separation between quantum and classical query complexity known to date—specifically, we define a black-box problem that requires exponentially many queries in the classical bounded-error case, but can be solved exactly in the quantum case with a single query (and a polynomial number of auxiliary operations). The problem is simple to define and the quantum algorithm solving it is also simple when described in terms of certain quantum Fourier transforms (QFTs) that have natural properties with respect to the algebraic structures of finite fields. These QFTs may be of independent interest, and we also investigate generalizations of them to noncommutative finite rings.

We analyze the Ohya-Masuda quantum algorithm that solves the so-called “satisfiability” problem, which is an NP-complete problem of the complexity theory. We distinguish three steps in the algorithm, and analyze the second step, in which a coherent superposition of states (a “pure” state) transforms into an “incoherent” mixture presented by a density matrix. We show that, if “nonideal” (in analogy with “nonideal” quantum measurement), this transformation can make the algorithm to fail in some cases. On this basis we give some general notions on the physical implementation of the Ohya-Masuda algorithm.

Quantum computation is a new computing model based on fundamental quantum mechanical principle. Grover's algorithm finds the solution for a searching problem in the square root time of exhaustive search. Brassard, Hoyer, Tapp's algorithm counts the number of solutions for a searching problem. Through exploiting the two quantum algorithms, we propose a quantum algorithm for solving a new cryptography problem -- polynomial root finding problem, which could be used to design a cryptosystem. The algorithm will take O(rootM/t) steps for finding one of the t solutions to the problem, where M is the modular of the equation. The success rate of the algorithm is a constant and the cost of the algorithm depends on the calculations of modular exponentiation and the number of iterations.

We report an experimental demonstration of a one-way implementation of a quantum algorithm solving Simon's Problem - a black box period-finding problem which has an exponential gap between the classical and quantum runtime. Using an all-optical setup and modifying the bases of single-qubit measurements on a five-qubit cluster state, key representative functions of the logical two-qubit version's black box can be queried and solved. To the best of our knowledge, this work represents the first experimental realization of the quantum algorithm solving Simon's Problem. The experimental results are in excellent agreement with the theoretical model, demonstrating the successful performance of the algorithm. With a view to scaling up to larger numbers of qubits, we analyze the resource requirements for an n-qubit version. This work helps highlight how one-way quantum computing provides a practical route to experimentally investigating the quantum-classical gap in the query complexity model., 9 pages, 5 figures

A quantum algorithm for solving the classical NP-complete problem - the Hamilton circuit is presented. The algorithm employs the quantum SAT and the quantum search. algorithms. The algorithm is square-root faster than classical algorithm, and becomes exponentially faster than classical algorithm if nonlinear quantum mechanical computer is used.

Besides the Hidden Subgroup Problem, the second large class of quantum speed-ups is for functions with constant-sized 1-certificates. This includes the OR function, solvable by the Grover algorithm, the element distinctness, the triangle and other problems. The usual way to solve them is by quantum walk on the Johnson graph. We propose a solution for the same problems using span programs. The span program is a computational model equivalent to the quantum query algorithm in its strength, and yet very different in its outfit. We prove the power of our approach by designing a quantum algorithm for the triangle problem with query complexity O(n35/27) that is better than O(n13/10) of the best previously known algorithm by Magniez et al.

Delay-constraint minimum cost multicast tree problem is a classical combinatorial optimization problem. It has been proved to be a NP-complete problem. Many algorithms have been proposed to solve the problem. This paper proposed a tree-based quantum optimization algorithm for the multicast tree problem. In this algorithm, the probability amplitude of each side was stored in a quantum, and the quantum evolved by quantum gate, each quantum moved in the direction of better solution. The evolutionary process merged the trees rather than paths directly to accelerate the convergence and improved the quality of solutions. Simulation results showed that the algorithm in this paper performed well at searching and convergence speed.

This paper investigates the computability of mathematical relations in a quantum query model. The important task in complexity theory is to find examples with a large gap between classical and quantum algorithm complexity of the same computational problem. We present new results in quantum query algorithm design that allow achieving a large separation between classical and quantum query complexity of a specific relation. We demonstrate an example where quantum query algorithm for a finite relation needs more than two times fewer queries than the best possible classical analogue. We also show that relation can be extended to infinite family of relations with an input of general size N.

Superposition principle of quantum theory is introduced in this paper. The genetic algorithm has been described by quantum superposition theory. It implies genetic algorithm is essentially a kind of quantum algorithm in the classical computer on the reduced order to achieve. It through genetic manipulation to input superposition states data into classical computer. In the evolutionary process, a lot of implicit modes have been measured, so as to realize the parallel search and processing in the model space. It is implicit parallelism of genetic algorithm. The quantum theory can also be used to analyze the problem?s computational complexity. Quantum superposition theory and entropy are used to get the problem?s lower bound of computational complexity. Using this method the problem?s lower bound of computational complexity is only decided by itself. It is independent of the algorithm?s details.

The Quantum Adiabatic Algorithm has been proposed as a general purpose algorithm for solving hard optimization problems on a quantum computer. Early work on very small sizes indicated that the running time (complexity) only increased as a (quite small) power of the problem size N. We report results of Quantum Monte Carlo simulations, using parallel tempering, with which we determine the minimum energy gap (and hence get information the complexity) for much bigger sizes than was possible before. The aim is to see if there is a “crossover” to exponential complexity at large N. We present data for the typical (median) complexity as a function of N, which indicate a crossover to a first order transition at large sizes. This implies that the complexity is exponential at large N, at least for the problem studied.

We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity., 10 pages, LaTeX, no figures, final version to appear in FOCS'98

In this paper we present a classical parallel quantum algorithm for the satisfiability problem. We have exploited the classical parallelism of quantum algorithms developed in [G.L. Long and L. Xiao, Phys. Rev. A 69 (2004) 052303], so that additional acceleration can be gained by using classical parallelism. The quantum algorithm first estimates the number of solutions using the quantum counting algorithm, and then by using the quantum searching algorithm, the explicit solutions are found.