  The search problem is to find a state satisfying certain properties out of a
given set. Grover's algorithm drives a quantum computer from a prepared initial
state to the target state and solves the problem quadratically faster than a
classical computer. The algorithm uses selective transformations to distinguish
the initial state and target state from other states. It does not succeed
unless the selective transformations are very close to phase-inversions. Here
we show a way to go beyond this limitation. An important application lies in
quantum error-correction, where the errors can cause the selective
transformations to deviate from phase-inversions. The algorithms presented here
are robust to errors as long as the errors are reproducible and reversible.
This particular class of systematic errors arise often from imperfections in
apparatus setup. Hence our algorithms offer a significant flexibility in the
physical implementation of quantum search.
