Playing Mastermind on quantum computers

Abstract

From the 1970s up to now, Mastermind, a classic two-player game, has attracted plenty of attention, not only from the public as a popular game, but also from the academic community as a scientific issue. Mastermind with n positions and k colors is formally described as: the codemaker privately chooses a secret $s\in [k]^n$, and the coderbreaker want to determine $s$ in as few queries like $f_s(x)$ as possible to the codemaker, where $f_s(x)$ indicates how x is close to s. The complexity of a strategy is measured by the number of queries used. In this work we study playing Mastermind on quantum computers in both non-adaptive and adaptive settings, obtaining efficient quantum algorithms which are all exact (i.e., return the correct result with certainty) and show huge quantum speedups. Technically, we develop a three-step framework for designing quantum algorithms for the general string learning problem, which not only allows huge quantum speedups on playing Mastermind, but also may shed light on exploring quantum speedups for other string learning problems.