Abstract: By adding a parameter θ in M. Jakobi et al’s protocol [Phys. Rev. A 83, 022301 (2011)], we present a flexible quantum-key-distribution-based protocol for quantum private queries. We show that, by adjusting the value of θ, the average number of the key bits Alice obtains can be located on any fixed value the users wanted for any database size. And the parameter k is generally smaller (even k = 1 can be achieved) when θ < π/4, which implies lower complexity of both quantum and classical communications. Furthermore, the users can choose a smaller θ to get better database security, or a larger θ to obtain a lower probability with which Bob can correctly guess the address of Alice’s query.

Abstract: We propose an adiabatic quantum algorithm for generating a quantum pure state encoding of the PageRank vector, the most widely used tool in ranking the relative importance of internet pages. We present extensive numerical simulations which provide evidence that this algorithm can prepare the quantum PageRank state in a time which, on average, scales polylogarithmically in the number of web pages. We argue that the main topological feature of the underlying web graph allowing for such a scaling is the out-degree distribution. The top-ranked log(n) entries of the quantum PageRank state can then be estimated with a polynomial quantum speed-up. Moreover, the quantum PageRank state can be used in "q-sampling" protocols for testing properties of distributions, which require exponentially fewer measurements than all classical schemes designed for the same task. This can be used to decide whether to run a classical update of the PageRank.

Abstract: In the context of evolutionary quantum computing in the literal meaning, a quantum crossover operation has not been introduced so far. Here, we introduce a novel quantum genetic algorithm which has a quantum crossover procedure performing crossovers among all chromosomes in parallel for each generation. A complexity analysis shows that a quadratic speedup is achieved over its classical counterpart in the dominant factor of the run time to handle each generation.

Abstract: We give two time- and space-efficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates Specifically, our simulations show that every language solvable by a bounded-error quantum algorithm running in time t and space s is also solvable by an unbounded-error randomized algorithm running in time O(t logt) and space O(s+ logt), as well as by a bounded-error quantum algorithm restricted to use an arbitrary universal set and running in time O(t polylogt) and space O(s+ logt), provided the universal set is closed under adjoint We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds As an application of our randomized simulation, we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability Our bound applies to MAJSAT and MAJMAJSAT, which are the problems of determining the truth

Abstract: Quantum computers are known to be qualitatively more powerful than classical computers, but so far only a small number of different algorithms have been discovered that actually use this potential. It would therefore be highly desirable to develop other types of quantum algorithms that widen the range of possible applications. Here we propose an efficient and exact quantum algorithm for finding the square-free part of a large integer - a problem for which no efficient classical algorithm exists. The algorithm relies on properties of Gauss sums and uses the quantum Fourier transform. We give an explicit quantum network for the algorithm. Our algorithm introduces new concepts and methods that have not been used in quantum information processing so far and may be applicable to a wider class of problems.

Abstract: We present new quantum algorithms for Boolean Matrix Multiplication in both the time complexity and the query complexity settings. As far as time complexity is concerned, our results show that the product of two n x n Boolean matrices can be computed on a quantum computer in time O(n3/2 + nl3/4), where l is the number of non-zero entries in the product, improving over the output-sensitive quantum algorithm by Buhrman and Spalek that runs in [EQUATION] time. This is done by constructing a quantum version of a recent algorithm by Lingas, using quantum techniques such as quantum counting to exploit the sparsity of the output matrix. As far as query complexity is concerned, our results improve over the quantum algorithm by Vassilevska Williams and Williams based on a reduction to the triangle finding problem. One of the main contributions leading to this improvement is the construction of a triangle finding quantum algorithm tailored especially for the tripartite graphs appearing in the reduction.

Abstract: Tremendous efforts have been paid for realization of fault-tolerant quantum computation so far However, preexisting fault-tolerant schemes assume that a lot of qubits live together in a single quantum system, which is incompatible with actual situations of experiment Here we propose a novel architecture for practically scalable quantum computation, where quantum computation is distributed over small-size (four-qubit) local systems, which are connected by quantum channels We show that the proposed architecture works even with the error probability 01% of local operations, which breaks through the consensus by one order of magnitude Furthermore, the fidelity of quantum channels can be very low $\sim$ 07, which substantially relaxes the difficulty of scaling-up the architecture All key elements and their accuracy required for the present architecture are within reach of current technology The present architecture allows us to achieve efficient scaling of quantum computer, as has been achieved in today's classical computer

Abstract: Quantum information science explores the frontier of highly complex quantum states, the "entanglement frontier". This study is motivated by the observation (widely believed but unproven) that classical systems cannot simulate highly entangled quantum systems efficiently, and we hope to hasten the day when well controlled quantum systems can perform tasks surpassing what can be done in the classical world. One way to achieve such "quantum supremacy" would be to run an algorithm on a quantum computer which solves a problem with a super-polynomial speedup relative to classical computers, but there may be other ways that can be achieved sooner, such as simulating exotic quantum states of strongly correlated matter. To operate a large scale quantum computer reliably we will need to overcome the debilitating effects of decoherence, which might be done using "standard" quantum hardware protected by quantum error-correcting codes, or by exploiting the nonabelian quantum statistics of anyons realized in solid state systems, or by combining both methods. Only by challenging the entanglement frontier will we learn whether Nature provides extravagant resources far beyond what the classical world would allow.

Abstract: Quantumcomputersareknowntobequalitativelymorepowerfulthanclassicalcomputers,butsofaronlya small number of different algorithms have been discovered that actually use this potential. It would therefore be highly desirable to develop other types of quantum algorithms that widen the range of possible applications.Hereweproposeanefficientandexactquantumalgorithmforfindingthesquare-freepartofa large integer - a problem for which no efficient classical algorithm exists. The algorithm relies on properties of Gauss sums and uses the quantum Fourier transform. We give an explicit quantum network for the algorithm. Our algorithm introduces new concepts and methods that have not been used in quantum information processing so far and may be applicable to a wider class of problems.

Abstract: This tutorial is the first part of a series of two articles on quantum computation In this first paper, we present the field of quantum computation from a broad perspective We review the mathematical background and informally discuss physical implementations of quantum computers Finally, we present the main primitives used in quantum algorithms

Abstract: We study the quantum walk search algorithm of Shenvi et al. (Phys Rev A 67:052307, 2003) on data structures of one to two spatial dimensions, on which the algorithm is thought to be less efficient than in three or more spatial dimensions. Our aim is to understand why the quantum algorithm is dimension dependent whereas the best classical algorithm is not, and to show in more detail how the efficiency of the quantum algorithm varies with spatial dimension or accessibility of the data. Our numerical results agree with the expected scaling in 2D of $$O(\sqrt{N \log N})$$ , and show how the prefactors display significant dependence on both the degree and symmetry of the graph. Specifically, we see, as expected, the prefactor of the time complexity dropping as the degree (connectivity) of the structure is increased.

Abstract: Topological quantum computing has recently proven itself to be a powerful computational model when constructing viable architectures for large scale computation. The topological model is constructed from the foundation of a error correction code, required to correct for inevitable hardware faults that will exist for a large scale quantum device. It is also a measurement based model of quantum computation, meaning that the quantum hardware is responsible \emph{only} for the construction of a large, computationally universal quantum state. This quantum state is then strategically consumed, allowing for the realisation of a fully error corrected quantum algorithm. The number of physical qubits needed by the quantum hardware and the amount of time required to implement an algorithm is dictated by the manner in which this universal quantum state is consumed. In this paper we examine the problem of algorithmic optimisation in the topological lattice and introduce the required elements that will be needed when designing a classical software package to compile and implement a large scale algorithm on a topological quantum computer.

Abstract: The simulation of quantum algorithms in classic computers is a task which requires high processing and storing capabilities, limiting the size of quantum systems supported by the simulators However, optimizations for reduction of temporal and spatial complexities are promising, expanding the capabilities of some simulators The main contribution of this work consists in designing optimizations resulting from the description of quantum transformations using Quantum Processes and Partial Quantum Processes conceived in the qGM theoretical model These processes, when computed on the VPE-qGM execution environment, require low execution time and result in the improvement of the performance, allowing the simulation of more complex quantum algorithms The performance evaluation of this proposal was performed by benchmarks used in similar works and included the sequential simulation of quantum algorithms up to 24 qubits The results are promising when compared to the state-of-art, indicating the possibilities of advances in this research

Abstract: A quantum algorithm is defined by a sequence of operations that runs on a realistic model of quantum computation. Since the first quantum algorithm proposed by David Deutsch(1985), a large number of impressive quantum algorithms have been developed. Quantum random walks on a graph, which are analogous to classical stochastic walk, form the basis for some of the recent quantum algorithms that promise to significantly outperform existing classical random walk algorithms. Though research has been done on the application of quantum random walk to important computational problems, very little work has been done on its quantum circuit design. There are two types of quantum random walk algorithms: discrete-time and continuous-time. In this paper, we propose quantum circuit designs for both types of random walk algorithms that operate on various graphs. We consider in detail two important problems to which random walk algorithms are applicable: the triangle finding problem and binary welded tree problem. Though there exist a few research works related to quantum circuit design for random walk on graphs, to the best of our knowledge, the circuit designs we present here are first of their kind. We also provide an estimate of the quantum cost of these circuits for several physical machine descriptions (PMDs)of quantum systems, based on the number of quantum operations and execution cycles.

Abstract: Quantum algorithms are sequences of abstract operations, per­ formed on non-existent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contribu­ tions of Abramsky, Goecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its clas­ sical interfaces. Intuitively, classical data can be recognized as just those data that can be manipulated using variables, i.e. copied, deleted, and abstracted over. A categorical framework of polynomial extensions provides a convenient language for specifying quantum algorithms, with a clearly distinguished clas­ sical fragment, familiar from functional programming. As a running example, we reconstruct Simon's algorithm, which is a sim­ ple predecessor of Shor's quantum algorithms for factoring and discrete loga­ rithms. The abstract specification in the framework of polynomial categories displays some of the fundamental program transformations involved in devel­ oping quantum algorithms, and points to the computational resources, whether quantum or classical, which are necessary for the various parts of the execution. The relevant resources are characterized as categorical structures. They are normally supported by the standard Hilbert space model of quantum mechan­ ics, but in some cases they can also be found in other, nonstandard models. We conclude the paper by sketching an implementation of Simon's algorithm using just abelian groups and relations.

Abstract: We describe a quantum algorithm for computing the intersection of two sets and its application to associative memory. The algorithm is based on a modification of Grover's quantum search algorithm (Grover, 1996). We present algorithms for pattern retrieval, pattern completion, and pattern correction. We show that the quantum associative memory can store an exponential number of memories and retrieve them in sub-exponential time. We prove that this model has advantages over known classical associative memories as well as previously proposed quantum models.

Abstract: Since the discovery of Shor's algorithm, the anxiety about quantum computation has increased. A large amount of research has been conducted to discover new algorithms and to build a quantum computer. But it seems that a general purpose quantum computer is far from being achieved. Meanwhile, cryptographers around the world started to look for security algorithms that resist to quantum attacks, but these still need improvement to achieve practical execution time. This work proposes a quantum-classical hybrid architecture, focusing on photonic quantum computers. A small quantum coprocessor implementing the Grover search algorithm is used to perform the search for roots of polynomials in $\mathbb{F}_{p^q}$. This coprocessor is used to accelerate the decoding process of the McEliece cryptosystem.

Abstract: Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.

Abstract: Two quantum search algorithms are proposed for known and unknown number of solutions. The first algorithm begins with an arbitrary rotation phase Grover search algorithm by recursive equations, then a sub-algorithm (G α algorithm) and the corresponding quantum circuits are designed, the probability of success and expected number of iterations of the sub-algorithm to find a solution are analyzed. Based on these results, we design the whole algorithm and estimate the expected number of iterations to search all solutions. The design of the second algorithm follows the same steps. We firstly study a quantum counting algorithm, then design a sub-algorithm (QCG algorithm), and analyze the probability of success and the expected number of iterations to find a solution. Finally the whole algorithm for all solutions is designed and analyzed. The analysis results show that these two algorithms find all solutions in the expected times of $O(t\sqrt{N})$ (t is a number of solutions and N is a total of states).

Abstract: The theory of decoherent histories is an attempt to derive classical physics from positing only quantum laws at the fundamental level without notions of a classical apparatus or collapse of the wave-function. Searching for a marked target in a list of N items requires \Omega(N) oracle queries when using a classical computer, while a quantum computer can accomplish the same task in O{\sqrt{N}} queries using Grover's quantum algorithm. We study a closed quantum system executing Grover algorithm in the framework of decoherent histories and find it to be an exactly solvable model, thus yielding an alternate derivation of Grover's famous result. We also subject the Grover-executing computer to a generic external influence without needing to know the specifics of the Hamiltonian insofar as the histories decohere. Depending on the amount of decoherence, which is captured in our model by a single parameter related to the amount of information obtained by the environment, the search time can range from quantum to classical. Thus, we identify a key effect induced by the environment that can adversely affect a quantum computer's performance and demonstrate exactly how classical computing can emerge from quantum laws.

Abstract: Discrete time quantum walks are known to be universal for quantum computation. This has been proven by showing that they can simulate a universal gate set. In this paper we examine computation in terms of language acceptance and present two ways in which discrete time quantum walks can accept some languages with certainty. These walks can take quantum as well as classical inputs, and we show that when the input is quantum, the walks can be interpreted as performing state discrimination.

Abstract: The focus of this study is on developing a framework for a Quantum Algorithm Processing Unit (QAPU) with the hybrid architecture for classical-quantum algorithms. The framework is used to increase the implementation performance of quantum algorithms and design Quantum Processing Units (QPU). The framework shows a general plan for the architecture of quantum processors who is capable of run the quantum algorithms. In particular, the QAPU can be used as a quantum node to design a quantum multicomputer. At first, the hybrid architecture is designed for the quantum algorithms. Then, the relationship between the classical and the quantum part of hybrid algorithms is extracted, and main stages of the hybrid algorithm are developed. Next, the framework of the QAPU is designed. Furthermore, the framework is implemented and simulated for the existing quantum algorithms on a classic computer. It is shown that the framework is appropriate for quantum algorithms.

Abstract: In order for finding a good individual for a given fitness function in the context of evolutionary computing, weintroduce a novel semiclassical quantum genetic algorithm. It has both of quantum crossover and quantum mutationprocedures unlike conventional quantum genetic algorithms. A complexity analysis shows a certain improvementover its classical counterpart. 1 Introduction Continuous development has been performed on genetic algorithms [1–3]. Along with the development of quantumcomputing [4, 5], quantum-inspired classical algorithms for evolutionary computing have been developed [6–13] (seealso a review [14] and references therein). In addition, classical genetic algorithms to evolve quantum circuits have alsobeen studied by several authors [15–29] (see also review articles [30, 31]). These algorithms are, however, designed towork on classical computers. Quantum genetic algorithms (QGAs), in its literal meaning, nonetheless, have gatheredcomparably little attention and a few works [32–36] have been performed so far. Evolutionary computing on quantumarchitectures will achieve more attention if there is a scenario to establish significant improvement over classicalcounterparts. Indeed, Malossini

Abstract: In the classical model of a computer, the most fundamental building block, the bit, can only exist in one of two distinct states, a 0 or a 1. Computations are carried out by logic gates that act on these bits to produce other bits. Unless there is duplicate (parallel) hardware, only one problem instance (i.e. input data set) can be handled at a time. In this classical computing, increasing the number of bits increases the complexity of the problem and the time necessary to arrive at a solution. A quantum algorithm consists of a sequence of operations on a register, to transform it into a state which, when measured, yields the desired result with high probability. An n-bit quantum register can store an exponential amount of information. This paper aims at taking advantage of the superiority of quantum computing over classical computing to solve the problem of searching unstructured databases for a particular item or more than one item in good time. The general aim of this work is to establish the correctness and optimality of Grover’s quantum database search algorithm compared against classical database search methods in order to investigate the superiority or otherwise of quantum computing over classical computing. This is followed by a simulation of the algorithm using a classical computer, namely through functions that are present in MATLAB, referred to as “Quantum Computing Functions”. Povzetek: Clanek predstavlja implementacijo kvantnega iskanja v nestrukturiranih bazah.

Abstract: A quantum bug, or "qbug", is the fundamental unit of problems in quantum key distribution. It can include a particular attack, an inaccurate use of the technology, a loophole of the theory or a hidden side channel. In this manuscript we detail one of them, related to the choice of the protocol and its security proof in the finite-size scenario. The treatment makes use of linear programming, a tool that well adapts to the practical constraints imposed by an actual quantum key distribution set up.

Abstract: Programming a quantum computer posts a challenge. It is not straight forward to transfer the current programming skill on a classical computer to a quantum computer. This work presents an example of programming a quantum computer. The compact genetic algorithm is used as a target as it is powerful and popular method in evolutionary computation. A quantum bit (qubit) concept was introduced as a basis for storing information. The representation of quantum register has benefits over classical computing, i.e. the quantum operation allows manipulating qubits in the way that it is impossible in a classical computer. This paper demonstrates the enhancement in terms of solution quality and speed introduced by quantum computation. The simulation of quantum computing is carried out for solving a problem using the compact genetic algorithm.

Abstract: Based on analysis on properties of quantum linear superposition, to overcome the complexity of existing quantum associative memory which was proposed by Ventura, a new storage method for multiply patterns is proposed in this paper by constructing the quantum array with the binary decision diagrams. Also, the adoption of the nonlinear search algorithm increases the pattern recalling speed of this model which has multiply patterns to $O( {\log_{2}}^{2^{n -t}} ) = O( n - t )$ time complexity, where n is the number of quantum bit and t is the quantum information of the t quantum bit. Results of case analysis show that the associative neural network model proposed in this paper based on quantum learning is much better and optimized than other researchers’ counterparts both in terms of avoiding the additional qubits or extraordinary initial operators, storing pattern and improving the recalling speed.

Abstract: A quantum learning machine for binary classification of qubit states that does not require quantum memory is introduced and shown to perform with the minimum error rate allowed by quantum mechanics for any size of the training set. This result is shown to be robust under (an arbitrary amount of) noise and under (statistical) variations in the composition of the training set, provided it is large enough. This machine can be used an arbitrary number of times without retraining. Its required classical memory grows only logarithmically with the number of training qubits, while its excess risk decreases as the inverse of this number, and twice as fast as the excess risk of an "estimate-and-discriminate" machine, which estimates the states of the training qubits and classifies the data qubit with a discrimination protocol tailored to the obtained estimates.