A new algorithm for fixed point quantum search

Abstract

The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase-$π/3$ search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from $ε$ to $ε^{2q+1}$ using $q$ oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-$π/3$ search algorithm but much better average-case behavior. Furthermore the new algorithm gives $ε^{2q+1}$ convergence for all integral $q$, whereas the Phase-$π/3$ search algorithm requires $q$ to be $(3^{n}-1)/2$ with $n$ a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.