t b a b a

Each iteration Q will rotate the system state by 2θ radians toward the solutions of the
searching problem. Thus after m iterations, the measurement on the final state Q m | ψ0〉 will
produce a “Good” state with probability equal to sin 2 ((2m + 1)θ). This algorithm Qsearch
finds a good solution using an expected number of applications of A and A −1 which are in
O( 1
√ a ). [BHM02, Ho00]
3.3 Extension of Grover’s Algorithm
Grover’s algorithm can find a solution in N items, requiring only O(N) operations. After
that, many special extensions of Grover’s algorithm were developed to address different
applications.
1. Fix-point Search: Grover’s algorithm is able to find a target state in an unsorted
database of size N in only O( √ N) queries. It is achieved by designing the iterative
transformations in a way that each iteration results in a small rotation of the state
vector in a two-dimensional Hilbert space that includes the target state. If the number
of iterative steps is perfectly chosen, the process will stop just at the target state. Oth-
erwise, it may overshoot the target. By replacing the selective inversions by selective
phase shifts of π/3, the algorithm preferentially converges to the target state irrespec-
tive of the step size or number of iterations. [Gro05] This feature leads to robust search
algorithms and also to new schemes for quantum control and error correction [RG05].
A different approach can be found in [TGP06].
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