A strong direct product theorem for quantum query complexity

Abstract

We show that quantum query complexity satisfies a strong direct product theorem. This means that computing $k$ copies of a function with less than $k$ times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in $k$. For a boolean function $f$ we also show an XOR lemma---computing the parity of $k$ copies of $f$ with less than $k$ times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, is always at least as large as the additive adversary method, which is known to characterize quantum query complexity.

V2: 19 pages (various additions and improvements, in particular: improved parameters in the main theorems due to a finer analysis of the output condition, and addition of an XOR lemma and a threshold direct product theorem in the boolean case). V3: 19 pages (added grant information)

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