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 ∈ [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) is called a black-peg query if f s (x) = B s (x) = |{i : s i = x i }|, and a black-white-peg query if f s (x) = {B s (x), W s (x)} with W s (x) = max σ∈Pn |{i : s i = x σ(i) }| − B s (x) and P n denoting the set of all permutations of {1, 2, . . . , n}. The complexity of a strategy is measured by the number of queries used. In this work we have a systematic study on quantum strategies for playing Mastermind, obtaining a full characterization of the quantum complexity and optimal quantum algorithms in both non-adaptive and adaptive settings.(i) The quantum complexity is proved to be Θ(k) in the non-adaptive setting. Two non-adaptive quantum algorithms are constructed for Mastermind with n positions and k colors that returns the secret with certainty: one with O(k log k)complexity and the other with O(k)-complexity. In addition, an algorithm with O(1) queries is constructed for the case k = 2. The algorithm-design skills in the three algorithms have substantial differences, and may be helpful for solving other problems. (ii) The quantum complexity is proved to be Θ( √ k) in the adaptive setting. An optimal adaptive quantum algorithm is constructed for Mastermind with n positions and k colors that uses O( √ k) queries and returns the secret with certainty.Our results show that the codebreaker wins greatly more on quantum computers than on classical computers. In the non-adaptive setting, when k ≤ n the classical complexity is Θ( n log k max{log(n/k),1} ) monotonically increasing with respect to n, and when k > n the current best classical algorithm has complexity O(k log k) which is * The authors are ordered alphabetically.
In this paper, we study for the first time how well quantum algorithms perform on matroid properties. We show that quadratic quantum speedup is possible for the calculation problem of finding the girth or the number of circuits (bases, flats, hyperplanes) of a matroid, and the decision problem of deciding whether a matroid is uniform or Eulerian, by giving a uniform lower bound Ω( n n/2 ) on the query complexity for all these problems. On the other hand, for the uniform matroid decision problem, we present an asymptotically optimal quantum algorithm which achieves the lower bound, and for the girth problem, we give an almost optimal quantum algorithm with query complexity O(log n n n/2 ). In addition, for the paving matroid decision problem, we prove a lower bound Ω( n n/2 /n) on the query complexity, and give an O( n n/2 ) quantum algorithm.
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