Quantum algorithm to distinguish Boolean functions of different weights

By the weight of a Boolean function $f$, denoted by $wt(f)$, we mean the number of inputs for which $f$ outputs 1. Given a promise that an $n$-variable Boolean function (available in the form of oracle) is of weight either $\lfloor{2^n\sin^2{\frac{t}{2t+1}\frac{\pi}{2}}}\rceil$ or $\lfloor{2^n\cos^2{\frac{t}{2t+1}\frac{\pi}{2}}}\rceil$ ($\lfloor{q}\rceil$ means the nearest integer corresponding to the real value $q$), we show that one can suitably apply Grover's operator for $t$-many iterations to decide which case this is with a probability almost unity for large $n$ and $t$ in $O(poly(n))$. On the other hand, the best known probabilistic classical algorithm has a success probability close to 0.5 (from above) after $t$ many steps when $t$ is large. We further show that the best known probabilistic classical algorithm can achieve a success probability almost unity only after $t^k$ many iterations where $k > 2$.This indicates a quadratic speed up on time complexity in the quantum domain with respect to the best known result in the classical domain. To the best of our knowledge, this is the first time Grover's quantum algorithm is being studied in depth in distinguishing Boolean functions of different weights which are very close to weight $2^{n-1}$ as $t$ increases. Further our detailed analysis gives a clear idea on the exact efficiency of the probabilistic classical algorithm