  The standard quantum search algorithm lacks a feature, enjoyed by many
classical algorithms, of having a fixed-point, i.e. a monotonic convergence
towards the solution. Here we present two variations of the quantum search
algorithm, which get around this limitation. The first replaces selective
inversions in the algorithm by selective phase shifts of $\frac{\pi}{3}$. The
second controls the selective inversion operations using two ancilla qubits,
and irreversible measurement operations on the ancilla qubits drive the
starting state towards the target state. Using $q$ oracle queries, these
variations reduce the probability of finding a non-target state from $\epsilon$
to $\epsilon^{2q+1}$, which is asymptotically optimal. Similar ideas can lead
to robust quantum algorithms, and provide conceptually new schemes for error
correction.

Quantum Physics