Phase Matching in Quantum Searching

Each iteration in Grover’s original quantum search algorithm contains 4 steps: two Hadamard-Walsh transformations and two amplitudes inversions. When the inversion of the marked state is replaced by arbitrary phase rotation θ and the inversion for the prepared state |γ〉 is replaced by rotation through φ, we found that these phase rotations must satisfy a matching condition θ = φ. Approximate formula for the amplitude of the marked state after an arbitrary number of iterations are also derived. When phase matching is obtained, we found that the generalized quantum search is still a rotation in 2-dimensional space, but with a small angle β ≈ (2 sin θ 2 )β, where θ is the angle of phase rotation of the marked state and 2β is the angle of U(2) rotation when the phase rotations are the inversions in the original version. We give also a simple explanation of the phase matching requirement. 03.67-a, 03.67.Lx, 03.65-w, 89.70+c