On the quantum and classical complexity of solving subtraction games

Abstract

We study algorithms for solving Subtraction games, which are sometimes referred as one-heap Nim games. We describe a quantum algorithm which is applicable to any game on DAG, and show that its query complexity for solving an arbitrary Subtraction game of n stones is O(n3/2log n). The best known deterministic algorithms for solving such games are based on the dynamic programming approach [8]. We show that this approach is asymptotically optimal and that classical query complexity for solving a Subtraction game Θ(n2) in general. Of course, this difference between classical and quantum algorithms is far from the best known examples, but, up to our knowledge, this paper is the first constructive “quantum” contribution to the algorithmic game theory.