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A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We established such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the disjointness function. These results imply a time-space tradeoff T²S = Ω(N³) for sorting N items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication.
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 Additional Information
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Citation:
Hartmut Klauck, Robert Spalek, Ronald de Wolf,
"Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs,"
focs,
pp. 12-21,
45th Annual IEEE Symposium on Foundations of Computer Science (FOCS'04),
2004
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