On the Hitting Times of Quantum Versus Random Walks

The hitting time of a classical random walk (Markov chain) is the time required to detect the presence of—or equivalently, to find—a marked state. The hitting time of a quantum walk is subtler to define; in particular, it is unknown whether the detection and finding problems have the same time complexity. In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Then, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem (Tulsi A.: Phys. Rev. A 78:012310 2008) for the 2D grid. Extending his result, we show that we can find a unique marked element with constant probability and with the same complexity as detection for a large class of quantum walks—the quantum analogue of state-transitive reversible ergodic Markov chains. Further, we prove that for any reversible ergodic Markov chain P, the quantum hitting time of the quantum analogue of P has the same order as the square root of the classical hitting time of P. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. In doing so, we define a notion of reversibility for a broad class of quantum walks and show how to derive from any such quantum walk a classical analogue. For the special case of quantum walks built on reflections, we show that the hitting time of the classical analogue is exactly the square of the quantum walk.     

Nonadaptive quantum query complexity

We study the power of nonadaptive quantum query algorithms, which are algorithms whose queries to the input do not depend on the result of previous queries. First, we show that any bounded-error nonadaptive quantum query algorithm that computes a total boolean function depending on n variables must make Ω(n) queries to the input in total. Second, we show that, if there exists a quantum algorithm that uses k nonadaptive oracle queries to learn which one of a set of m boolean functions it has been given, there exists a nonadaptive classical algorithm using queries to solve the same problem. Thus, in the nonadaptive setting, quantum algorithms for these tasks can achieve at most a very limited speed-up over classical query algorithms.     

Efficient and exact quantum compression

We present a divide and conquer based algorithm for optimal quantum compression/decompression, using O(n(log⁴n)log log n) elementary quantum operations. Our result provides the first quasi-linear time algorithm for asymptotically optimal (in size and fidelity) quantum compression and decompression. We also outline the quantum gate array model to bring about this compression in a quantum computer. Our method uses various classical algorithmic tools to significantly improve the bound from the previous best known bound of O(n³) for this operation.     

Unbounded-error quantum query complexity

This work studies the quantum query complexity of Boolean functions in an unbounded-error scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded-error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, connecting the query and communication complexity results, we show that the “black-box” approach to convert quantum query algorithms into communication protocols by Buhrman-Cleve—Wigderson [STOC’98] is optimal even in the unbounded-error setting.We also study a related setting, called the weakly unbounded-error setting, where the cost of a query algorithm is given by q+log(1/2(p−1/2)), where q is the number of queries made and p>1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight multiplicative Θ(logn) separation between quantum and classical query complexity in this setting for a partial Boolean function. The asymptotic equivalence between them is also shown for some well-studied total Boolean functions.     

Quantum novel genetic algorithm based on parallel subpopulation computing and its application

A quantum novel genetic algorithm based on subpopulation parallel computing is presented, where quantum coding and rotation angle are improved to inspire more efficient genetic computing methods. In the algorithm, each axis of the solution space is divided into k parts, the individual (or chromosome) from each different subspace being coded differently, and the probability amplitude of each quantum bit or Q-bit is regarded as two paratactic genes. The basic quantum computing theory and classical quantum genetic algorithm are briefly introduced before a novel algorithm is presented for the function optimum and PID problem. Through a comparison between the novel algorithm and the classical counterpart, it is shown that the quantum inspired genetic algorithm performs better on running speed and optimization capability.     

Classical and superposed learning for quantum weightless neural networks

A supervised learning algorithm for quantum neural networks (QNN) based on a novel quantum neuron node implemented as a very simple quantum circuit is proposed and investigated. In contrast to the QNN published in the literature, the proposed model can perform both quantum learning and simulate the classical models. This is partly due to the neural model used elsewhere which has weights and non-linear activations functions. Here a quantum weightless neural network model is proposed as a quantisation of the classical weightless neural networks (WNN). The theoretical and practical results on WNN can be inherited by these quantum weightless neural networks (qWNN). In the quantum learning algorithm proposed here patterns of the training set are presented concurrently in superposition. This superposition-based learning algorithm (SLA) has computational cost polynomial on the number of patterns in the training set.     

An all-pair quantum SVM approach for big data multiclass classification

In this paper, we discuss a quantum approach for the all-pair multiclass classification problem. In an all-pair approach, there is one binary classification problem for each pair of classes, and so there are k(k???1)/2 classifiers for a k-class classification problem. As compared to the classical multiclass support vector machine that can be implemented with polynomial run time complexity, our approach exhibits exponential speedup due to quantum computing. The quantum all-pair algorithm can also be used with other classification algorithms, and a speedup gain can be achieved as compared to their classical counterparts.     

Quantum speed-up for unsupervised learning

We show how the quantum paradigm can be used to speed up unsupervised learning algorithms. More precisely, we explain how it is possible to accelerate learning algorithms by quantizing some of their subroutines. Quantization refers to the process that partially or totally converts a classical algorithm to its quantum counterpart in order to improve performance. In particular, we give quantized versions of clustering via minimum spanning tree, divisive clustering and k-medians that are faster than their classical analogues. We also describe a distributed version of k-medians that allows the participants to save on the global communication cost of the protocol compared to the classical version. Finally, we design quantum algorithms for the construction of a neighbourhood graph, outlier detection as well as smart initialization of the cluster centres.     

A Random Base Change Algorithm for Permutation Groups

A new random base change algorithm is presented for a permutation group G acting on n points whose worst case asymptotic running time is better for groups with a small to moderate size base than any known deterministic algorithm. To achieve this time bound, the algorithm requires a random generator Rand(G) producing a random element of G with the uniform distribution and so that the time for each call to Rand (G) is bounded by some function f(n, G). The random base change algorithm has probability 1 - 1/|G| of completing in time O(f(n, G) log |G|) and outputting a data structure for representing the point stabilizer sequence relative to the new ordering. The algorithm requires O(n log |G|) space and the data structure produced can be used to test group membership in time O(n log |G|). Since the output of this algorithm is a data structure allowing generation of random group elements in time O(n log |G|), repeated application of the random base change algorithms for different orderings of the permutation domain of G will always run in time O (n log² |G|). An earlier version of this work appeared in Cooperman and Finkelstein (1992b).     

改进的多目标元素量子搜索算法

Grover量子搜索算法解决了未加排序的数据库搜索问题,在2ⁿ个元素中搜索M个目标元素,其计算复杂度为O（（2ⁿ/M）^-2）,相对于经典算法实现了二次加速,但是,当目标元素个数接近2ⁿ/2时该算法成功率只达到50%。从任意相位的Grover变换从发,给出一种改进的多目标元素量子搜索算法,该算法在目标元素个数M≥2ⁿ/4时,只用一次Grover变换就能以概率1完成搜索。     

多目标元素的量子搜索算法

Grover量子搜索算法解决了未加整理的数据库搜索问题,在2ⁿ个元素中搜索M个目标元素时,计算复杂度为O（√2ⁿ/M）,相对于经典算法实现了二次加速,但Grover算法在目标元素个数接近2ⁿ/2时成功率较低。提出了一种针对多目标元素的量子搜索算法,当目标元素个数大于2ⁿ/3时,能以不低于97.36%的概率找到目标元素。     

Generalized Ramsey numbers through adiabatic quantum optimization

Ramsey theory is an active research area in combinatorics whose central theme is the emergence of order in large disordered structures, with Ramsey numbers marking the threshold at which this order first appears. For generalized Ramsey numbers r(G, H), the emergent order is characterized by graphs G and H. In this paper we: (i) present a quantum algorithm for computing generalized Ramsey numbers by reformulating the computation as a combinatorial optimization problem which is solved using adiabatic quantum optimization; and (ii) determine the Ramsey numbers \(r({\mathscr {T}}_{m},{\mathscr {T}}_{n})\) for trees of order \(m,n = 6,7,8\), most of which were previously unknown.     

The on-line first-fit algorithm for radio frequency assignment problems

The radio frequency assignment problem is to minimize the number of frequencies used by transmitters with no interference in radio communication networks; it can be modeled as the minimum vertex coloring problem on unit disk graphs. In this paper, we consider the on-line first-fit algorithm for the problem and show that the competitive ratio of the algorithm for the unit disk graph G with χ(G)=2 is 3, where χ(G) is the chromatic number of G. Moreover, the competitive ratio of the algorithm for the unit disk graph G with χ(G)>2 is at least 4−3/χ(G). The average performance for the algorithm is also discussed in this paper.     

Quantum Relief algorithm

Relief algorithm is a feature selection algorithm used in binary classification proposed by Kira and Rendell, and its computational complexity remarkably increases with both the scale of samples and the number of features. In order to reduce the complexity, a quantum feature selection algorithm based on Relief algorithm, also called quantum Relief algorithm, is proposed. In the algorithm, all features of each sample are superposed by a certain quantum state through the CMP and rotation operations, then the swap test and measurement are applied on this state to get the similarity between two samples. After that, Near-hit and Near-miss are obtained by calculating the maximal similarity, and further applied to update the feature weight vector WT to get \({\overline{WT}}\) that determine the relevant features with the threshold \(\tau \). In order to verify our algorithm, a simulation experiment based on IBM Q with a simple example is performed. Efficiency analysis shows the computational complexity of our proposed algorithm is O(M), while the complexity of the original Relief algorithm is O(NM), where N is the number of features for each sample, and M is the size of the sample set. Obviously, our quantum Relief algorithm has superior acceleration than the classical one.     

Enhanced algorithms for Local Search

Let G=(V,E) be a finite graph, and be any function. The Local Search problem consists in finding a local minimum of the function f on G, that is a vertex v such that f(v) is not larger than the value of f on the neighbors of v in G. In this note, we first prove a separation theorem slightly stronger than the one of Gilbert, Hutchinson and Tarjan for graphs of constant genus. This result allows us to enhance a previously known deterministic algorithm for Local Search with query complexity , so that we obtain a deterministic query complexity of , where n is the size of G, d is its maximum degree, and g is its genus. We also give a quantum version of our algorithm, whose query complexity is of . Our deterministic and quantum algorithms have query complexities respectively smaller than the algorithm Randomized Steepest Descent of Aldous and Quantum Steepest Descent of Aaronson for large classes of graphs, including graphs of bounded genus and planar graphs.     

Conflict-free star-access in parallel memory systems

We study conflict-free data distribution schemes in parallel memories in multiprocessor system architectures. Given a host graph G, the problem is to map the nodes of G into memory modules such that any instance of a template type T in G can be accessed without memory conflicts. A conflict occurs if two or more nodes of T are mapped to the same memory module. The mapping algorithm should: (i) be fast in terms of data access (possibly mapping each node in constant time); (ii) minimize the required number of memory modules for accessing any instance in G of the given template type; and (iii) guarantee load balancing on the modules. In this paper, we consider conflict-free access to star templates, i.e., to any node of G along with all of its neighbors. Such a template type arises in many classical algorithms like breadth-first search in a graph, message broadcasting in networks, and nearest neighbor based approximation in numerical computation. We consider the star-template access problem on two specific host graphs—tori and hypercubes—that are also popular interconnection network topologies. The proposed conflict-free mappings on these graphs are fast, use an optimal or provably good number of memory modules, and guarantee load balancing.     

Algorithms for regular and irregular coulomb and bessel functions

Algorithms for computing Coulomb-Bessel functions are considered, with emphasis on obtaining accurate values when the argument x is inside the classical turning point x_λ. Algorithms of Barnett et al. for the generalized Coulomb functions and their derivatives are discussed in the context of the phase integral formalism. Modified or alternative algorithms are considered that are designed to be valid for all values of argument x and index λ for the functions F_λ(x), G_λ(x). An algorithm for a ccelerating convergence of a power series by conversion to a continued fraction is presented, and is applied to the evaluation of spherical Bessel functions. An explicit formula for the integrand of the phase integral is presented for spherical Bessel functions. The methods considered need to be augmented by an efficient algorithm for computing the logarithmic derivative of G₀ + iF₀ for Coulomb functions when x is smaller than the charge parameter η.     

A Polycyclic Quotient Algorithm

This paper describes a generalization of the Gröbner basis method to the integral group ring of a polycyclic group. A polycyclic quotient algorithm is developed using this method. SupposeGis a group given by a finite presentation andG⁽ⁿ⁾is thenth term in the derived series ofG. A polycyclic quotient algorithm computes the quotientG/G⁽ⁿ⁾if it is polycyclic. An implementation of this algorithm in C has been developed and its efficiency is encouraging.     

Feedback vertex sets in star graphs

In a graph G=(V,E), a subset F⊂V(G) is a feedback vertex set of G if the subgraph induced by V(G)?F is acyclic. In this paper, we propose an algorithm for finding a small feedback vertex set of a star graph. Indeed, our algorithm can derive an upper bound to the size of the feedback vertex set for star graphs. Also by applying the properties of regular graphs, a lower bound can easily be achieved for star graphs.     

A fast distributed approximation algorithm for minimum spanning trees

We present a distributed algorithm that constructs an O(log n)-approximate minimum spanning tree (MST) in any arbitrary network. This algorithm runs in time Õ(D(G) + L(G, w)) where L(G, w) is a parameter called the local shortest path diameter and D(G) is the (unweighted) diameter of the graph. Our algorithm is existentially optimal (up to polylogarithmic factors), i.e., there exist graphs which need Ω(D(G) + L(G, w)) time to compute an H-approximation to the MST for any $H\,\in\,[1, \Theta({\rm log} n)]We present a distributed algorithm that constructs an O(log n)-approximate minimum spanning tree (MST) in any arbitrary network. This algorithm runs in time ?(D(G) + L(G, w)) where L(G, w) is a parameter called the local shortest path diameter and D(G) is the (unweighted) diameter of the graph. Our algorithm is existentially optimal (up to polylogarithmic factors), i.e., there exist graphs which need Ω(D(G) + L(G, w)) time to compute an H-approximation to the MST for any . Our result also shows that there can be a significant time gap between exact and approximate MST computation: there exists graphs in which the running time of our approximation algorithm is exponentially faster than the time-optimal distributed algorithm that computes the MST. Finally, we show that our algorithm can be used to find an approximate MST in wireless networks and in random weighted networks in almost optimal ?(D(G)) time.