Theorem: QFT_2 = (1/sqrt(2))*[[1,1],[1,-1]] coincides with Hadamard; unitary check = I; det = -1

Theorem (QFT_2 = Hadamard coincidence): for N = 2, omega = exp(2*pi*i/2) = e^{i*pi} = -1. The 2x2 QFT matrix element is <j|QFT_2|k> = (-1)^{jk}/sqrt(2), giving QFT_2 = (1/sqrt(2)) * [[1, 1], [1, -1]] - identical to the Hadamard gate H.…