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

In the 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 quantum algorithms in this domain are parity decision trees, in which the parity of two bits can be obtained using a single query. Thus, exact quantum query algorithms outperforming parity decision trees are rare. In this paper we explore a class of $\Omega \left( 2^{\frac{\sqrt{n}}{2}} \right)$ non-symmetric Boolean functions that we design based on Direct Sum Constructions. The (classical) Deterministic Query Complexity $D(f)$ of all functions in this class is $n$. We design a family of exact quantum query algorithms for this class of functions that require $\lfloor \frac{3n}{4} \rfloor$ queries and we show that and our family of algorithms is optimal, outperforming any possible generalized parity decision tree technique. The generalized parity decision tree model is a stronger version of the parity decision tree model in which parity of any $i \leq n$ bits can be obtained in a single query. For example, we can show a class of function for which using our strategy requires $\lfloor \frac{3n}{4} \rfloor$ queries, compared to $n-1$ in generalized parity decision tree technique. We achieve this separation by designing a new generic exact quantum algorithm that is based on analyzing the $\mathbb{F}_2$ polynomial of a function and un-entangling multiple qubits in a single query whose states are dependent on input values, which gives us the advantage for the said classes of functions. To the best of our knowledge, this is the first family of algorithms beyond generalized parity (and thus parity) for a general class of non-symmetric functions.