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 effectiveness of the proposed approach.Comment: 19 pages, 5 figures, 5 table

Quantum State Tomography and Quantum Games

We develop a technique for single qubit quantum state tomography using the mathematical setup of generalized quantization scheme for games. In our technique Alice sends an unknown pure quantum state to Bob who appends it with |0><0| and then applies the unitary operators on the appended quantum state and finds the payoffs for Alice and himself. It is shown that for a particular set of unitary operators these elements become equal to Stokes parameters for an unknown quantum state. In this way an unknown quantum state can be measured and reconstructed. Strictly speaking this technique is not a game as no strategic competitions are involved.Comment: 9 pages, 3 figure

Analysis of two-player quantum games in an EPR setting using geometric algebra

The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of Clifford geometric algebra (GA). In this setting, the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, which is then obtained as proper subset of the corresponding quantum game. As examples, using GA we analyze the games of Prisoners' Dilemma and Stag Hunt when played in the EPR type setting.Comment: 20 pages, no figure, revise

A probabilistic approach to quantum Bayesian games of incomplete information

A Bayesian game is a game of incomplete information in which the rules of the game are not fully known to all players. We consider the Bayesian game of Battle of Sexes that has several Bayesian Nash equilibria and investigate its outcome when the underlying probability set is obtained from generalized Einstein-Podolsky-Rosen experiments. We find that this probability set, which may become non-factorizable, results in a unique Bayesian Nash equilibrium of the game.Comment: 18 pages, 2 figures, accepted for publication in Quantum Information Processin

N-player quantum games in an EPR setting

The $N$-player quantum game is analyzed in the context of an Einstein-Podolsky-Rosen (EPR) experiment. In this setting, a player's strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players' strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for $N$-qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the mixed Nash equilibrium and the payoffs are determined. Players' payoffs are then defined with linear functions so that common two-player games can be easily extended to the $N$-player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners' Dilemma game for general $N \ge 2$. We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs.Comment: 26 pages, 2 figure

A Review on Data Clustering Algorithms for Mixed Data

Clustering is the unsupervised classification of patterns into groups (clusters). The clustering problem has been addressed in many contexts and by researchers in many disciplines; this reflects its broad appeal and usefulness as one of the steps in exploratory data analysis. In general, clustering is a method of dividing the data into groups of similar objects. One of significant research areas in data mining is to develop methods to modernize knowledge by using the existing knowledge, since it can generally augment mining efficiency, especially for very bulky database. Data mining uncovers hidden, previously unknown, and potentially useful information from large amounts of data. This paper presents a general survey of various clustering algorithms. In addition, the paper also describes the efficiency of Self-Organized Map (SOM) algorithm in enhancing the mixed data clustering

Percolation Theories for Quantum Networks

Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network's indirect connectivity. This realization leads to the emergence of an alternative theory called ``concurrence percolation,'' which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design