Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs

Abstract

A strong direct product theorem says that if we want to computek independent instances of a function, using less than k timesthe resources needed for one instance, then our overall successprobability will be exponentially small in k.We establish such theorems for the classical as well as quantumquery complexity of the OR function. This implies slightlyweaker direct product results for all total functions.We prove a similar result for quantum communicationprotocols computing k instances of the Disjointness function. Our direct product theorems imply a time-space tradeoffT^2 S= Omega(N^3) for sorting N items on a quantum computer, whichis optimal up to polylog factors. They also give several tighttime-space and communication-space tradeoffs for the problems ofBoolean matrix-vector multiplication and matrix multiplication.<br/