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Abstract : |
We present several results related to quantum algorithms with very small error probability or even with zero-error. Firstly, we give a tight analysis of the trade-offs between the number of queries of quantum search algorithms, their error probability, the size of the search space, and the number of solutions in the search space. Secondly, we use this to show that a quantum computer cannot amplify the success probability of a given bounded-error algorithm A significantly faster than a classical computer (provided it only uses A as a black-box). Thirdly, we prove nearly optimal separations between quantum and classical computers for monotone Boolean functions in the zero-error model. Fourthly, we obtain the first total relation with a quantum-classical gap in the zero-error model of communication complexity. Finally, we prove separations for monotone graph properties in the zero-error and other error models. This implies that the evasiveness conjecture for such properties does not hold for quantum computers. 1 Motivation and Statement of Results A general goal in the theory of randomized algorithms is to obtain fast algorithms with small error probabilities. Along these lines is also the goal of obtaining fast algorithms that are zero-error, |
