Performance of equal phase-shift search for one iteration
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Grover presented the phase-shift search by replacing the selective inversions by selective phase shifts of π/3. In this paper, we investigate the phase-shift search with general equal phase shifts. We show that for small uncertainties, the failure probability of the phase-π/3 search is smaller than the general phase-shift search and for large uncertainties, the success probability of the large phase-shift search is larger than the phase-π/3 search. Therefore, the large phase-shift search is suitable for large-size of databases.
PACS.03.67.Lx Quantum computation
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- L.K. Grover et al., e-print arXiv:quant-ph/0603132 Google Scholar
- In the reviewer's report on this paper, the reviewer investigated fixed points of the phase-θ search algorithm as follows. The reviewer let f(epsilonn)=epsilonn+1=epsilon n[1+2(cosθ-1)(1-epsilonn)]2. Then the reviewer showed that the equation has 0 to be an attractive fixed point when 0<θ<π/2 because 0<f′(0)<1, cosθ/(cosθ-1) to be an attractive fixed point when π/2<θ<2π/3 because |f′ (cosθ/(cosθ-1))|<1, and all the fixed points to be repulsive when 2π/3<θ<π. Following the reviewer's idea, we can show when θ=π/2, 0 is a semi-attractive fixed point because f′(0)=1 and f′′(0)=-8 and when θ=2π/3, 1/3 is also a semi-attractive fixed-point because f′(1/3)=-1 and f′′(1/3)=-6 Google Scholar