Quantum Reachability Games

The reachability of quantum games refers to the ability of two players to reach a certain subspace of Hilbert spaces in a quantum game modeled by a newly proposed quantum game process (QGP). We refer to this as quantum reachability games (QRGs). Reachability analysis is crucial in studying the strategic behavior and decision-making processes of players in quantum games, as it aids in devising optimal strategies to achieve desired objectives. In a QRG, players make decisions based on their quantum strategies related to decision epochs, which are represented by quantum operations or quantum measurements. We present an algorithm for evaluating two-player strategies, and investigate the optimal QRG for two players, where the goal of player 1 is to maximize his/her probability of reaching a given target subspace, while player 2 aims to minimize it. We show that any optimal QRG are determined, i.e., both of the two players have the same probability of winning in non-cooperative reachability games. Furthermore, our results also demonstrate that reachability games in the quantum world can be undecidable in certain cases. These findings are significantly distinct from those in the probabilistic or classical scenarios. Finally, we present three case studies, penny flipover, ambushing quantum walk and peer-to-peer scheme game on quantum network, to illustrate the potential applications and implications of QRGs.