Enormous successes have been made by quantum algorithms during the last decade. In this paper, we combine the quantum game with the problem of data clustering, and then develop a quantum-game-based clustering algorithm, in which data points in a dataset are considered as players who can make decisions and implement quantum strategies in quantum games. After each round of a quantum game, each player's expected payoff is calculated. Later, he uses a link-removing-and-rewiring (LRR) function to change his neighbors and adjust the strength of links connecting to them in order to maximize his payoff. Further, algorithms are discussed and analyzed in two cases of strategies, two payoff matrixes and two LRR functions. Consequently, the simulation results have demonstrated that data points in datasets are clustered reasonably and efficiently, and the clustering algorithms have fast rates of convergence. Moreover, the comparison with other algorithms also provides an indication of the effectiveness of the proposed approach.
Deep Dive into A Novel Clustering Algorithm Based on Quantum Games.
Enormous successes have been made by quantum algorithms during the last decade. In this paper, we combine the quantum game with the problem of data clustering, and then develop a quantum-game-based clustering algorithm, in which data points in a dataset are considered as players who can make decisions and implement quantum strategies in quantum games. After each round of a quantum game, each player’s expected payoff is calculated. Later, he uses a link-removing-and-rewiring (LRR) function to change his neighbors and adjust the strength of links connecting to them in order to maximize his payoff. Further, algorithms are discussed and analyzed in two cases of strategies, two payoff matrixes and two LRR functions. Consequently, the simulation results have demonstrated that data points in datasets are clustered reasonably and efficiently, and the clustering algorithms have fast rates of convergence. Moreover, the comparison with other algorithms also provides an indication of the effective
Quantum computation is an extremely exciting and rapidly growing field. More recently, an increasing number of researchers with different backgrounds, ranging from physics, computer sciences and information theory to mathematics and philosophy, are involved in researching properties of quantum-based computation [1]. During the last decade, a series of significant breakthroughs had been made. One was that in 1994 Peter Shor surprised the world by proposing a polynomial-time quantum algorithm for integer factorization [2], while in the classical world the best-known classical factoring algorithm works in superpolynomial time. Three years later, in 1997, Lov Grover proved that a quantum computer could search an unsorted database in the square root of the time [3]. Meanwhile, Gilles Brassard et al. combined ideas from Grover's and Shor's quantum algorithms to propose a quantum counting algorithm [4].
In recent years, many interests focus on the quantum game theory and considerable work has been done. For instance, D. A. Meyer [5] studied the Penny Flip game in the quantum world firstly. His result showed that if a player was allowed to implement quantum strategies, he would always defeat his opponent who played the classical strategies and increase his expected payoff as well. J. Eisert et al. [6] quantized the Prisoners’ Dilemma and demonstrated that the dilemma could be escaped when both players resort to quantum strategies. A. P. Flitney et al. [7] generalized Eisert’s result, the miracle move, i.e., the result of the game would move towards the quantum player’s preferred result, while the other player used classical strategies. L. Marinatto et al. [8] investigated the Battle of the Sexes game in quantum domain. Their result showed that there existed a unique equilibrium in the game, when the entangled strategies were allowed. C. F. Lee et al. [9] reported that the quantum game is more efficient than the classical game, and they found an upper bound for this efficiency. Besides, some experiments about the quantum games have also been implemented on different quantum computers [10,11,12]. For more details about quantum games, see [13].
Successes achieved by quantum algorithms make us guess that powerful quantum computers can figure out solutions faster and better than the best known classical counterparts for certain types of problems. Furthermore, it is more important that they offer a new way to find potentially dramatic algorithmic speed-ups. Therefore, we may ask naturally: can we construct quantum versions of classical algorithms or present new quantum algorithms to solve the problems in pattern recognition faster and better on a quantum computer? Following this idea, some researchers have proposed their novel methods and demonstrated exciting results [14,15,16,17,18].
In addition, data clustering is a main branch of Pattern Recognition, which is widely used in many fields such as pattern analysis, data mining, information retrieval and image segmentation. In these fields, however, there is usually little priori knowledge available about the data. In response to these restrictions, clustering methodology come into being which is particularly suitable for the exploration of interrelationships among data points. Data clustering is the formal study of algorithms and methods for grouping or classifying unlabeled data points [19]. In other words, its task is to find the inherent structure of a given collection of unlabeled data points and group them into meaningful clusters [19]. In this paper, we attempt to combine the quantum game with the problem of data clustering in order to establish a novel clustering algorithm based on quantum games. In our algorithms, unlabeled data points in a dataset are regarded as players who can make decisions in quantum games. On a timevarying network formed by players, each player is permitted to use quantum strategies and plays a 2 × 2 entangled quantum game against every one of his neighbors respectively. Later, he applies a link-removing-and-rewiring (LRR) function to remove the links of neighbors with small payoffs and create new links to neighbors with higher payoffs at the same time. Furthermore, the strength of links between a player and his neighbors is different from one another, which is updated by the Grover iteration. During quantum games, the structure of network and the strength of links between players tend toward stability gradually. Finally, if each player only connects to the neighbor with the highest strength, the network will naturally divide into several separate parts, each of which corresponds to a cluster. The remainder of this paper is organized as follows: Section 2 introduces some important concepts about the quantum computation and the quantum Prisoners’ Dilemma briefly. In Section 3, the algorithms are established in two cases of strategies, payoff matrices and link-removing-and-rewiring (LRR) functions, and then they are elaborated and analyzed
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