Unbounded-error quantum query complexity

This work studies the quantum query complexity of Boolean functions in an unbounded-error scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded-error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, connecting the query and communication complexity results, we show that the “black-box” approach to convert quantum query algorithms into communication protocols by Buhrman–Cleve—Wigderson [STOC’98] is optimal even in the unbounded-error setting.We also study a related setting, called the weakly unbounded-error setting, where the cost of a query algorithm is given by q+log(1/2(p−1/2)), where q is the number of queries made and p>1/2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight multiplicative Θ(logn) separation between quantum and classical query complexity in this setting for a partial Boolean function. The asymptotic equivalence between them is also shown for some well-studied total Boolean functions.