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.

💡 Research Summary

The paper introduces a novel clustering framework that treats each data point as a player in a quantum game. By allowing players to adopt quantum strategies and repeatedly engage in pairwise games with their current neighbors, the algorithm computes an expected payoff for every player after each round. Players then modify their local network topology through a “link‑removing‑and‑rewiring” (LRR) mechanism: the connection to the neighbor yielding the lowest payoff is dropped, and a new link is formed with a candidate neighbor that promises a higher payoff. The strength of each link is also adjusted proportionally to the associated payoff, creating a dynamic weighted graph that evolves toward a configuration where players are surrounded by high‑payoff partners.

Two families of quantum strategies are examined. The first uses classical‑style operators (identity and Pauli‑X), while the second employs genuinely quantum operations such as the Hadamard gate and controlled‑NOT, which generate superposition and entanglement. Correspondingly, two payoff matrices are defined: a cooperation‑oriented matrix that rewards mutual cooperation and a competition‑oriented matrix that favours defection. The LRR function is implemented in two variants – a deterministic threshold‑based rule and a stochastic selection rule that samples new neighbors according to a probability distribution derived from payoffs. By crossing the two strategy sets, two payoff matrices, and two LRR variants, the authors obtain four algorithmic configurations.

The algorithm proceeds as follows: (1) initialise a k‑nearest‑neighbour graph from the raw data; (2) for each player, play a quantum game with every neighbour, compute the expected payoff using the chosen matrix; (3) apply the LRR rule to prune the worst link and rewire to a better neighbour; (4) update link weights; (5) repeat until convergence, defined as a negligible change in average payoff across successive rounds. Convergence is typically reached within 10–15 iterations, regardless of dataset size.

Experimental evaluation uses synthetic datasets (well‑separated circles, moons, and imbalanced clusters) and several UCI benchmark sets (Iris, Wine, Glass, Breast‑Cancer). Performance metrics include clustering accuracy, Adjusted Rand Index, silhouette coefficient, and runtime. The configuration that combines quantum superposition strategies with the stochastic LRR rule consistently outperforms classical baselines such as k‑means, DBSCAN, and spectral clustering, achieving silhouette scores above 0.90 and accuracy improvements of 10–20 %. Moreover, the number of iterations required for convergence remains almost constant as the number of points grows from a few hundred to ten thousand, indicating good scalability in terms of iteration count, though the overall computational cost is higher due to the O(N²) neighbour‑search inherent in the LRR step.

The authors discuss several strengths of the approach: (i) dynamic neighbourhood adaptation helps escape local minima that plague static‑graph clustering methods; (ii) quantum strategies enlarge the strategic space, allowing players to explore configurations unavailable to purely classical algorithms; (iii) fast convergence suggests that the payoff‑driven rewiring quickly aligns the graph with the underlying cluster structure. They also acknowledge limitations: the algorithm’s performance depends on hyper‑parameters such as initial entanglement angle, neighbourhood size, and LRR thresholds; the current implementation relies on classical simulation of quantum games, so real‑world quantum hardware would introduce decoherence and gate‑error considerations; and the quadratic complexity of neighbour selection may become prohibitive for truly massive datasets without further optimisation.

Future work is outlined in four directions: (1) implementing the game on actual quantum processors to assess hardware‑induced noise effects; (2) developing adaptive schemes (e.g., reinforcement‑learning‑based tuning) for the quantum‑game parameters; (3) integrating approximate nearest‑neighbour structures (e.g., locality‑sensitive hashing) to reduce the LRR search cost; and (4) exploring hybrid models that combine the quantum‑game layer with conventional deep‑learning embeddings for high‑dimensional data.

In summary, the paper demonstrates that embedding quantum‑game dynamics into the clustering process yields a flexible, payoff‑driven graph evolution that can achieve superior clustering quality and rapid convergence compared with traditional methods. It opens a new interdisciplinary avenue where concepts from quantum information theory directly inform unsupervised learning algorithms.