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**Deveno** i find that hard to read (i know very little about quantum mechanics or computing) but it seems to me what is happening is this:$\displaystyle \left|\sum_b \omega^{(x_0 + rb)y}\right| = \left|\sum_b \omega^{x_0y}\omega^{rby}\right|$and since x_{0}y is independent of b:$\displaystyle = |\omega^{x_0y}|\left|\sum_b \omega^{rby}\right| = \left|\sum_b \omega^{rby}\right|$since ω lies on the unit circle, and thus any power of it has magnitude 1 (whereas the sum of such unit vectors may not lie on the unit circle).