Abstract: With current technologies, it seems to be very difficult to implement quantum computers with many qubits. It is therefore of importance to simulate quantum algorithms and circuits on the existing computers. However, for a large-size problem, the simulation often requires more computational power than is available from sequential processing. Therefore, simulation methods for parallel processors are required.We have developed a general-purpose simulator for quantum algorithms/ circuits on the parallel computer (Sun Enterprise4500). It can simulate algorithms/circuits with up-to 30 qubits. In order to test efficiency of our proposed methods, we have simulated Shor's factorization algorithm and Grover's database search, and we have analyzed robustness of the corresponding quantum circuits in the presence of both decoherence and operational errors. The corresponding results, statistics and analyses are presented in this paper.

Abstract: We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j-k with k>1. For the Wiener measure occurring in many applications we have k=2. We want to compute an e-approximation to path integrals whose integrands are at least Lipschitz. We prove: • Path integration on a quantum computer is tractable. • Path integration on a quantum computer can be solved roughly e-1 times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. • The number of quantum queries needed to solve path integration is roughly the square root of the number of function values needed on a classical computer using randomization. More precisely, the number of quantum queries is at most 4.46 e-1. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved. • The number of qubits is polynomial in e-1. Furthermore, for the Wiener measure the degree is 2 for Lipschitz functions, and the degree is 1 for smoother integrands. PACS: 03.67.Lx; 31.15Kb; 31.15.-p; 02.70.-c

Abstract: . 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.

Abstract: We pursue the possible connections between classical games and quantum computation. The Parrondo game is one in which a random combination of two losing games produces a winning game. We introduce novel realizations of this Parrondo effect in which the player can `win' via random reflections and rotations of the state-vector, and connect these to known quantum algorithms.

Abstract: We consider the problem of identifying a base k string given an oracle which returns information about the number of correct components in a query, specifically, the Hamming distance between the query and the solution, modulo r = max{2, 6 − k}. Classically this problem requires Ω(n logrk) queries. For k ∈ {2, 3, 4}, we construct quantum algorithms requiring only a single quantum query. For k > 4, we show that O(√k) quantum queries suffice. In both cases the quantum algorithms are optimal. PACS: 03.67.Lx

Abstract: It is known that quantum computer is more powerful than classical computer. In this paper we present quantum algorithms for some famous NP problems in graph theory and combination theory, these quantum algorithms are at least quadratically faster than the classical ones.

Abstract: The discovery of polynomial time prime factorization, secure key distribution, and fast database search algorithm have recently made quantum computing the most rapidly expanding research field. For a quantum algorithm to be useful, it is essential that the algorithm can be implemented using quantum circuits. Nanotechnology, in particular quantum mechanics based devices, can be used to realize such an algorithm. In this paper, we show how quantum Boolean circuits can be used to implement the oracle circuit and the inversion-about-average function in Grover's search algorithm. We also show that a slight modification of the oracle circuit can be used to search multiple targets.

Abstract: An application of an evolutionary approach to hardware design is presented. A genetic algorithm was developed to discover good designs for quantum computer algorithms. The algorithms are expressed as quantum operator sequences applied in a circuit model. The circuits discovered are configurations of special purpose quantum computers. We have been exploring the evolution of algorithms as alternative configurations of hardware. By simulation it is established that the circuits will correctly compute a small collection of basic, low-level functions. Experiments produced designs for primitive quantum computers that evaluate logical or arithmetic functions with a total of twelve or fewer inputs and outputs.

Abstract: We examine the effect of previous history on starting a computation on a quantum computer. Specifically, we assume that the quantum register has some unknown state on it, and it is required that this state be cleared and replaced by a specific superposition state without any phase uncertainty, as needed by quantum algorithms. We show that, in general, this task is computationally impossible.

Abstract: Quantum neural computation is a new paradigm based on the combination of classical neural computation and quantum theory. In this paper, the emergence reasons and characteristics of quantum neural computation are discussed first, then some typical computational models are introduced in detail, at the same time, several solutions to existing problems in these models are put forward. Finally, other related questions are also discussed.

Abstract: A quantum robot is a mobile quantum system, including an on board quantum computer and needed ancillary systems, that interacts with an environment of quantum systems. Quantum robots carry out tasks whose goals include making specified changes in the state of the environment or carrying out measurements on the environment. The environments considered so far, oracles, data bases, and quantum registers, are seen to be special cases of environments considered here. It is also seen that a quantum robot should include a quantum computer and cannot be simply a multistate head. A model of quantum robots and their interactions is discussed in which each task, as a sequence of alternating computation and action phases,is described by a unitary single time step operator T {approx} T{sub a} + T{sub c} (discrete space and time are assumed). The overall system dynamics is described as a sum over paths of completed computation (T{sub c}) and action (T{sub a}) phases. A simple example of a task, measuring the distance between the quantum robot and a particle on a 1D lattice with quantum phase path dispersion present, is analyzed. A decision diagram for the task is presented and analyzed.

Abstract: This paper demonstrates an application of an evolutionary approach for solving a class of non-trivial, hardware-design problems. Array processing features of the computer language APL simplify the implementation of an evolutionary solution in which simulation is performed by a genetic algorithm on a population of candidate solutions until one or more are satisfactory quantum algorithms. The objective of the simulation model is the automatic discovery of quantum computer algorithms. The algorithms are expressed in a circuit model that specifies the sequences in which quantum operators are to be applied. The automatically configured circuits operate as quantum computers that for the present have their domains of application limited to evaluation of only a few specific functions. The simulations use a small collection of basic and relatively low-level operators to obtain perfect results for five different target functions. These functions were chosen to demonstrate how, on a personal computer, an evolutionary method is able to discover novel designs of computing hardware. The results establish that it is feasible to use genetic algorithms in an evolutionary method to invent correct hardware designs. The progress of simulated evolution is directed by input/output constraints and a fitness function. Alternative configurations of circuit models can obtain algorithms with promise for future quantum computers. Current simulations yield primitive quantum algorithms with a total of twelve or fewer inputs and outputs. The discovered algorithms produce the correct results for evaluating the five selected logical and arithmetic functions.

Abstract: Since simulating quantum computers requires exponentially more classical resources, efficient algorithms are extremely helpful. We analyze algorithms that create single qubit and specific controlled qubit matrix representations of gates. Additionally, we use the simulator to investigate errors based on different probability distributions and to investigate the robustness of different 2-qubit multiplier circuits in the presence of operational errors.

Abstract: In this paper we consider the following question: how many bits of classical communication and shared random bits are necessary to simulate a quantum protocol involving Alice and Bob where they share k entangled quantum bits and do not communicate at all We prove that 2^k classical bits are necessary, even if the classical protocol is allowed an \epsilon chance of failure

Abstract: It appears, in principle, that the laws of quantum mechanics allow a quantum computer to solve certain mathematical problems more rapidly than can be done using a classical computer. However, in order to build such a quantum computer, a number of technological problems need to be overcome. A stepping stone to this goal is the implementation of relatively simple quantum algorithms using current experimental techniques. The research work presented in this thesis consists of several theoretical studies exploring small scale quantum algorithms and methods of implementing them. Included in this thesis are an investigation of a small scale version of the phase estimation algorithm, methods of implementing the quantum random walk, a discussion of protecting quantum information by encoding it in an oscillator, and a look at the power of a quantum computer with a restricted number of qubits.

Abstract: The time complexity of Grover's quantum searching algorithm is O( N ), but the algorithm's time complexity on classical computers is O( N ). It proves the strong power of quantum computers, and has important influence on quantum algorithm research. But we find that Grover's algorithm still has some problems. When the solution is not unique, this algorithm may cause failure. In this paper, we improve this quantum searching algorithm, and prove the correctness and validity of this improved searching algorithm. The improved algorithm can find solutions more efficiently than the original Grover's quantum searching algorithm.

Abstract: Although a great deal of research effort is currently focused on the technology and hardware of quantum computers, the design and development of algorithms for quantum computing also present an equivalent research challenge. In this paper we propose an adaptable design method for quantum algorithms based on superpositional permutation searching. We illustrate the use of this method with quantum algorithms for sorting and route finding problems.

Abstract: Arthur does not have a lot of time to spend performing difficult computations. He's recently obtained a quantum computer, but often it seems not to help - he only has a few quantum algorithms, and Merlin maintains that there aren't any other interesting ones, so Merlin is forced to convince the untrusting Arthur of the truth of various facts. However, Arthur and Merlin have a new resource at their disposal: quantum information. Some relationships among complexity classes defined by quantum Arthur-Merlin games and other commonly studied complexity classes are known, but many open questions remain. In this paper, I discuss quantum Arthur-Merlin games in detail, with an emphasis on open problems.

Abstract: The discovery of quantum error correction has greatly improved the long-term prospects for quantum computing technology. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment, or due to imperfect implementations of quantum logical operations. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. In principle, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. It may be possible to incorporate intrinsic fault tolerance into the design of quantum computing hardware, perhaps by invoking topological Aharonov-Bohm interactions to process quantum information.

Abstract: In the past decade, quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests that they may also speed up the simulation of some classical systems. I describe one class of discrete quantum algorithms which do so – quantum lattice–gas automata – and show how to implement them efficiently on standard quantum computers.

Abstract: We show how to perform a quantum search for a classical object, specifically for a classical object which performs no coherent evolution on the quantum computer being used for the search We do so by using interaction free measurement as a subroutine in a quantum search algorithm In addition to providing a simple example of how non-unitary processes which approximate unitary ones can be useful in a quantum algorithm, our procedure requires only one photon regardless of the size of the database, thereby establishing an upper bound on the amount of energy required to search an arbitrarily large database Alternatively, our result can be interpreted as showing how to perform an interaction free measurement with a single photon on an arbitrarily large number of possible bomb positions simultaneously We also provide a simple example demonstrating that in terms of the number of database queries, the procedure outlined here can outperform the best classical one