Classical simulation of one-query quantum distinguishers

We study the relative advantage of classical and quantum distinguishers of bounded query complexity over n -bit strings, focusing on the case of a single quantum query. A construction of Aaronson and Ambainis (STOC 2015) yields a pair of distributions that is ε -distinguishable by a one-query quantum algorithm, but O ( εk/ √ n )-indistinguishable by any non-adaptive k -query classical algorithm. We show that every pair of distributions that is ε -distinguishable by a one-query quantum algorithm is distinguishable with k classical queries and (1) advantage min { Ω( ε p k/n )) , Ω( ε 2 k 2 /n ) } non-adaptively (i.e., in one round), and (2) advantage Ω( ε 2 k/ √ n log n ) in two rounds. As part of our analysis, we introduce a general method for converting unbiased estimators into distinguishers.