  The Quantum Fourier Transform is a famous example in quantum computing for
being the first demonstration of a useful algorithm in which a quantum computer
is exponentially faster than a classical computer. However when giving an
explanation of the speed up, understanding computational complexity of a
classical calculation has to be taken on faith. Moreover, the explanation also
comes with the caveat that the current classical calculations might be
improved. In this paper we present a quantum computer code for a Galton Board
Simulator that is exponentially faster than a classical calculation using an
example that can be intuitively understood without requiring an understanding
of computational complexity. We demonstrate a straight forward implementation
on a quantum computer, using only three types of quantum gate, which calculates
$2^n$ trajectories using $\mathcal{O} (n^2)$ resources. The circuit presented
here also benefits from having a lower depth than previous Quantum Galton
Boards, and in addition, we show that it can be extended to a universal
statistical simulator which is achieved by removing pegs and altering the
left-right ratio for each peg.
