A strong direct product theorem for disjointness
Proceedings of the forty-second ACM symposium on Theory of computing
Optimal direct sum results for deterministic and randomized decision tree complexity
Information Processing Letters
Strong direct product theorems for quantum communication and query complexity
Proceedings of the forty-third annual ACM symposium on Theory of computing
The quantum query complexity of certification
Quantum Information & Computation
Improved output-sensitive quantum algorithms for Boolean matrix multiplication
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
The multiparty communication complexity of set disjointness
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Making polynomials robust to noise
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Communication lower bounds using directional derivatives
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Dual lower bounds for approximate degree and markov-bernstein inequalities
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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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 establish 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. Our direct product theorems imply a time-space tradeoff $T^2S=\Om{N^3}$ 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.