An Exact Quantum Query Algorithm Beyond Parity using Maiorana-McFarland (MM) type Bent functions

The Exact Quantum Query model is the least explored query model, and almost all of the functions for which non-trivial query algorithms exist are symmetric in nature. In this paper we explore the Maiorana-McFarland(MM) Bent functions, defined on all even $n$ variables. The Deterministic Query Complexity ($D(f)$) of all functions in this class is $n$. In this regard we construct a $\frac{n}{2} + \lceil \frac{n}{8} \rceil$ query exact quantum algorithm that is not a parity decision tree and evaluates a non-symmetric subclass of MM type Bent functions on $n$ variables consisting of $\left(2^{\lfloor \frac{n}{4} \rfloor}!\right)^2 2^{2^{\lfloor \frac{n}{4} \rfloor}}$ functions. Finally we also extend this technique for functions defined on odd values of $n$ using the Bent concatenation method. To the best of our knowledge, this is the first algorithm beyond parity for a general class of non-symmetric functions.