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Grover's quantum search algorithm evolves a quantum system from a knownsource state $|s\rangle$ to an unknown target state $|t\rangle$ using theselective phase inversions, $I_{s}$ and $I_{t}$, of these two states. In one ofthe generalizations of Grover's algorithm, $I_{s}$ is replaced by a generaldiffusion operator $D_{s}$ having $|s\rangle$ as an eigenstate and $I_{t}$ isreplaced by a general selective phase rotation $I_{t}^{\phi}$. A fast quantumsearch 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 $\ell \neq s$ hasto 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 introducingappropriate modifications of the algorithm. This allows access to a moregeneral class of diffusion operators for fast quantum search.

On the class of diffusion operators for fast quantum search