Electronic Colloquium on Computational Complexity
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TR15-175 | 5th November 2015 23:14

Separations in query complexity using cheat sheets


Authors: Scott Aaronson, Shalev Ben-David, Robin Kothari
Publication: 5th November 2015 23:28
Downloads: 590


We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al., to be shown in a unified manner.

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