# W. Thus, the action of the oracle on the superposition |s ) is to change the sign of the amplitude of the component /w) in s), namely: 14)=Uwls) See, for instance, ,http://en.wikipedia.org/wiki/Grover's_algorithm with notations consistent with those used in this section. The description here is inspired from the tutorial paper: C. Lavor, L.R.U. Manssur, and R. Portugal, Grover's algorithm: quantum database search, (2003), which can be downloaded from http://arxiv.org/abs/quant-ph/0301079. 19.4 Grover quantum database search algorithm (19.36) We can also rewrite the result in Eq. (19.36) in the form: Uw/s) (19.37) We can then interpret the action of the oracle U of the state |s) as a small rotation through an angle O. Indeed, we have cos = (s/y) (19.38) (the angle O being small, if one assumes that the database size, N, is large). Next we introduce a second operator Us according to (19.39) and define the Grover operator as G = UsUw. Let us see next the action of this operator on /s), which we shall refer to as a Grover iteration. From Eq. (19.37), we have: = Gs = UsUw/s) Us/V) = Usis) (2/s) (19.40) = the We observe from the result in Eq. (19.40) that the action of G on s) is to increase the amplitude of the eigenstate component /w), again by rotation. The corresponding Quantum database search algorithms rotation angle O' is given by (19.41) and according to Eq. (19.38), which shows that the rotation angles 0, O' due to U w and G are equal in absolute value. It is easy to show that /v/> = Uw/s) and /V/1 = Gss are, in fact, rotating by that same absolute angle O = O' in opposite directions away from s), forming an angle 0" = 20. Indeed, from Eqs. (19.37) and (19.40) we obtain: 1 = cos 20. (19.42) Finally, it is straightforward from the definitions in Eq. (19.40) that the cosine projection of 1/1/1 over the target state /w) takes the value (w/V/1) = (3 - 4/N)/VN, to compare with the initial projection (w/s) = 1/VN. The angle relations of the different states s), 1/1/1), and /w/ and relevant projections (w/s), (w/V1