Quantum combinatorial games

2026-07-07

AI Summary

This study investigates whether quantum mechanisms can confer an advantage to players in two-player, perfect-information games devoid of randomness. By formally integrating combinatorial game theory into a quantum framework—leveraging Zermelo’s theorem and finite deterministic adversarial models—the work reconstructs the theoretical foundations of such games. The analysis demonstrates that a quantum player gains no advantage against an opponent employing a perfectly optimal classical strategy; however, when the classical opponent exhibits suboptimal or biased strategies, quantum strategies can significantly enhance winning probabilities. These findings reveal that quantum effects possess practical utility only in non-equilibrium or imperfect adversarial settings, thereby establishing new theoretical boundaries and application perspectives for quantum game theory.

Abstract

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.

Problem

• quantum combinatorial games
• quantum advantage
• classical strategy
• perfect information
• adversarial models

Innovation

• quantum combinatorial games
• Zermelo's theorem
• quantum advantage
• perfect information games
• adversarial models