Robin Kothari: Separations in Query Complexity Using Cheat Sheets

Abstract: 2015 has been an exciting year for query complexity. I'll survey some of the breakthroughs ​made this year and then talk about some very recent work with my collaborators Scott Aaronson and Shalev Ben-David.​

We establish several new total function separations in query complexity. 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.