G q r = G q Q/r and therefore why G ( ψ q $_{r,l}$) ≈ G ( ψ q Q/r, $_{0}$) . The QFT operator takes each periodic state in the ascending branch to a corresponding periodic state in the descending branch that has the same entanglement, and vice versa (recall that the QFT operator is its own inverse). Finally, we present numerical evidence to support the claim that the QFT does not change the entanglement of periodic states (for sufficiently large q ). To this end we define the change of the Groverian entanglement of a state ψ induced by the QFT We examine this difference for periodic states as well as for random states. The random states are taken from a uniform distribution on the 2 q -dimensional complex unit sphere. The Groverian measure of random states of q qubits exhibits a distribution that was calculated before for certain values of q [24]. Any unitary operator U (such as the QFT), when applied to a sample of these states, will produce states whose Groverian measure exhibits the same distribution. Looking at values of Δ G ( ψ ) , for some states it is positive and for other states it is negative. The average of Δ G ( ψ ) over the random states is zero, but its distribution exhibits a certain width. In Fig. 3 we show the average of the absolute value of Δ G ( ψ ) , which is an estimate of the width of the distribution of the values of Δ G ( ψ ) . For periodic states | Δ G ( ψ ) | quickly decreases to zero with increasing q [Fig. 3(b)], while for random states it changes only slightly [Fig. 3(a)]. This demonstrates the special feature of periodic states, namely, that their Groverian measure is not affected by the QFT. Δ G ( ψ ) = G [QFT( ψ )] − G ( ψ ) . (33) The goal of the field of quantum algorithms is not only to find quantum algorithms that present a speedup over classical ones, but also to establish a deep understanding as to how this speedup is made possible. Currently, the best insight we can offer is that it is made possible by the combination of quantum superposition and quantum interference [30, 31]. Quantum superposition allows a sort of parallel computation, as all the states in the superposition go through the same unitary evolution in an independent manner. Since the results of each of the states cannot be accessed directly, we cannot fully exploit quantum parallelism. Nevertheless, through quantum interference the different parallel paths can VI. DISCUSSION 16