Strong Direct Product Theorems for Quantum Communication and Query Complexity

Abstract

A strong direct product theorem (SDPT) states that solving $n$ instances of a problem requires $\Omega(n)$ times the resources for a single instance, even to achieve success probability $2^{-\epsilon n}$ for a small enough constant $\epsilon>0.$ We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In both models, we prove the corresponding XOR lemmas and threshold direct product theorems.