  Understanding quantum speed-up over classical computing is fundamental for
the development of efficient quantum algorithms. In this paper, we study such
problem within the framework of the Quantum Query Model, which represents the
probability of output $x \in \{0,1\}^n$ as a function $\pi(x)$. We present a
classical simulation for output probabilities $\pi$, whose error depends on the
Fourier $1$-norm of $\pi$. Such dependence implies upper-bounds for the
quotient between the number of queries applied by an optimal classical
algorithm and our quantum algorithm, respectively. These upper-bounds show a
strong relation between Fourier $1$-norm and quantum parallelism. We show
applications to query complexity.
