This paper develops and analyses a novel quantum combinatorial game: quantum checkers (named Cheqqers). The concepts of superposition, entanglement, measurements and interference from quantum mechanics are integrated into the game of checkers by adding new types of legal moves. The addition of these new rules is done gradually by introducing several levels of 'quantumness'. Quantum checkers provides a framework for interpolating between a known and solved classical game and a more complex quantum game, and serves as 1) a benchmark for AI players learning to play quantum games and 2) an interesting game for human players that allows them to build intuition for quantum phenomena. We provide the initial analysis on the complexity of this game using random agents and a Monte Carlo tree search agent.
This paper develops and analyses a novel quantum combinatorial game: quantum checkers (named Cheqqers). The concepts of superposition, entanglement, measurements and interference from quantum mechanics are integrated into the game of checkers by adding new types of legal moves. The addition of these new rules is done gradually by introducing several levels of 'quantumness'. Quantum checkers provides a framework for interpolating between a known and solved classical game and a more complex quantum game, and serves as 1) a benchmark for AI players learning to play quantum games and 2) an interesting game for human players that allows them to build intuition for quantum phenomena. We provide the initial analysis on the complexity of this game using random agents and a Monte Carlo tree search agent.
Quantum phenomena have remained largely inaccessible to the general public. This can be attributed to the fact that we do not experience quantum mechanics on a tangible level in our daily lives. Games can provide an environment in which people can experience the strange behavior of the quantum world in a fun and mentally engaging way. Games could also offer an interesting test bed for near term quantum devices because they can be tailored to support varying amounts of quantum behavior through simple rule changes, which can be useful when dealing with limited resources. This paper explores the design of a variant of Chess which is built on top of unitary dynamics and includes non-trivial quantum effects such as superposition, entanglement, and interference. A movement ruleset is introduced which allows players to create superposition, entanglement, and interference effects. A measurement rule is also introduced which helps limit the size of the superposition, so the game remains tractable for a classical computer, and also helps make the game more comprehensible for a player.
The strategic Go game, known for the tedious mathematical complexities, has been used as a theme in many fiction, movies, and books. Here, we introduce the Go game and provide a new version of quantum Go in which the boxes are initially in a superposition of quantum states |0> and |1> and the players have two kinds of moves (classical and quantum) to mark each box. The mark on each box depends on the state to which the qubit collapses after the measurement. All other rules remain the same, except for here, we capture only one stone and not chains. Due to the enormous power and exponential speed-up of quantum computers as compared to classical computers, we may think of quantum computing as the future. So, here we provide a tangible introduction to superposition, collapse, and entanglement via our version of quantum Go. Finally, we compare the classical complexity with the quantum complexity involved in playing the Go game.
With the development of quantum computing and quantum information science, quantum games are emerging as a novel educational tool that helps students intuitively understand complex quantum concepts. This paper explores the application of quantum games in quantum education, with a focus on analyzing the rules, teaching design, and their impact on students' cognitive development in “Quantum Chess”. Quantum Chess integrates core concepts such as quantum superposition, entanglement, and measurement, allowing students to experience the unique phenomena of quantum physics during the game. In the future, further optimization of the integration of quantum games and education will be necessary to meet diverse learning needs and promote the broader application of quantum computing technology in the field of education.
Recently, a standardized framework was proposed for introducing quantum-inspired moves in mathematical games with perfect information and no chance. The beauty of quantum games-succinct in representation, rich in structures, explosive in complexity, dazzling for visualization, and sophisticated for strategic reasoning-has drawn us to play concrete games full of subtleties and to characterize abstract properties pertinent to complexity consequence. Going beyond individual games, we explore the tractability of quantum combinatorial games as whole, and address fundamental questions including: Quantum Leap in Complexity: Are there polynomial-time solvable games whose quantum extensions are intractable? Quantum Collapses in Complexity: Are there PSPACE-complete games whose quantum extensions fall to the lower levels of the polynomial-time hierarchy? Quantumness Matters: How do outcome classes and strategies change under quantum moves? Under what conditions doesn't quantumness matter? PSPACE Barrier for Quantum Leap: Can quantum moves launch PSPACE games into outer polynomial space We show that quantum moves not only enrich the game structure, but also impact their computational complexity. In settling some of these basic questions, we characterize both the powers and limitations of quantum moves as well as the superposition of game configurations that they create. Our constructive proofs-both on the leap of complexity in concrete Quantum Nim and Quantum Undirected Geography and on the continuous collapses, in the quantum setting, of complexity in abstract PSPACE-complete games to each level of the polynomial-time hierarchy-illustrate the striking computational landscape over quantum games and highlight surprising turns with unexpected quantum impact. Our studies also enable us to identify several elegant open questions fundamental to quantum combinatorial game theory (QCGT).
A combinatorial game is a deterministic game with no hidden information played between two opponents such as tic-tac-toe, checkers or chess. In this paper we extend combinatorial games to the quantum setting, by first revisiting and reformulating existing theory of classical combinatorial games. We investigate in which case a quantum opponent has an advantage over a classical one. Surprisingly, our instantiation of Zermelo's classical theorem in the quantum setting shows that the effects of quantum mechanics do not convey an advantage against a classical player that plays a perfect classical strategy. In a more realistic scenario, when the classical player makes mistakes, we show how the quantum opponent can amplify the mistake to increase their chance of winning. Our theory has application beyond the mere playing of board games and can be used as a tool in finite deterministic adversarial models with perfect information.
Quantum games have demonstrated to propose a new point of view for the solution of the classical problems and dilemmas in game theory. It has been shown in many literatures that quantum games are more efficient than classical games and provide a greater and more efficient solution. As interest in multi-player quantum games grow, methods are proposed to analyze and obtain efficient solutions for these problems. In quantum games with cooperation, analysis so far have taken into consideration all possible scenarios for the multiple players playing different strategies and in the calculation of payoff, setting the payoff of scenarios not possible to be zero. Chapter 5 presents the analysis of multiple players (three players and four players) using this method of analysis. In Chapter 6, a modification of the analysis was done. In Chapter 5, it should be noted that the probabilities of participating scenarios do not sum to unity which is technically not ideal. Hence Chapter 6 addresses this by performing re-normalizing the probabilities of the participating scenarios so that the probabilities sum to unity. The analysis was done for three and four player games and a comparison was made between the un-normalized method and the re-normalized method. It was found that re-normalization provides a more accurate representation of the classical game and hence can be a better solution for the quantum game.