Exact Quantum Query Algorithms Outperforming Parity -- Beyond The Symmetric functions

Abstract: In Exact Quantum Query model, almost all of the Boolean functions for which non-trivial query algorithms exist are symmetric in nature. The most well known techniques in this domain exploit parity decision trees, in which the parity of two bits can be obtained by a single query. Thus, exact quantum query algorithms outperforming parity decision trees are rare. In this paper we first obtain optimal exact quantum query algorithms ($Q_{algo}(f)$) for a direct sum based class of $\Omega \left( 2^{\frac{\sqrt{n}}{2}} \right)$ non-symmetric functions. We construct these algorithms by analyzing the algebraic normal form together with a novel untangling strategy. Next we obtain the generalized parity decision tree complexity ($D_{\oplus}(f)$) analysing the Walsh Spectrum. Finally, we show that query complexity of $Q_{algo}$ is $\lceil \frac{3n}{4} \rceil$ whereas $D_{\oplus}(f)$ varies between $n-1$ and $\lceil \frac{3n}{4} \rceil+1$ for different classes, underlining linear separation between the two measures in many cases. To the best of our knowledge, this is the first family of algorithms beyond generalized parity (and thus parity) for a large class of non-symmetric functions. We also implement these techniques for a larger (doubly exponential in $\frac{n}{4}$) class of Maiorana-McFarland type functions, but could only obtain partial results using similar algorithmic techniques.