On the class of diffusion operators for fast quantum search

Grover's quantum search algorithm evolves a quantum system from a known source state $|s\rangle$ to an unknown target state $|t\rangle$ using the selective phase inversions, $I_{s}$ and $I_{t}$, of these two states. In one of the generalizations of Grover's algorithm, $I_{s}$ is replaced by a general diffusion operator $D_{s}$ having $|s\rangle$ as an eigenstate and $I_{t}$ is replaced by a general selective phase rotation $I_{t}^{\phi}$. A fast quantum search is possible as long as the operator $D_{s}$ and the angle $\phi$ satisfies certain conditions. These conditions are very restrictive in nature. Specifically, suppose $|\ell\rangle$ denote the eigenstates of $D_{s}$ corresponding to the eigenphases $\theta_{\ell}$. Then the sum of the terms $|\langle \ell|t\rangle|^{2}\cot(\theta_{\ell}/2)$ over all $\ell \neq s$ has to be almost equal to $\cot(\phi/2)$ for a fast quantum search. In this paper, we show that this condition can be significantly relaxed by introducing appropriate modifications of the algorithm. This allows access to a more general class of diffusion operators for fast quantum search.