Imperfect-Information Games on Quantum Computers: A Case Study in Skat

We demonstrate how quantum computing can provide a promising framework for addressing imperfect information games, using the popular German card game Skat as a case study. Our approach employs quantum registers to encode the game's hidden and public information, along with tailored quantum gates to model the game's progress while respecting its rules. To evaluate player decisions, we introduce a score operator that projects the quantum state onto the winning subspace, enabling the estimation of winning probabilities via quantum algorithms such as quantum counting. This allows the efficient exploration of alternative strategies and potential improvements in computational speed compared to classical methods. By maximizing the payoff function, our framework yields actionable recommendations for optimal play. As classical computation faces severe limitations due to the exponential complexity of the game tree, we discuss the specific structural properties of Skat that may give rise to a quantum advantage once the problem size surpasses classical feasibility.