On the class of diffusion operators for fast quantum search

TL;DR

This paper broadens the class of diffusion operators usable in fast quantum search algorithms by relaxing previous restrictive conditions, enabling more flexible and efficient quantum search implementations.

Contribution

It introduces modifications to the quantum search algorithm that relax the restrictive conditions on diffusion operators, expanding the range of operators suitable for fast quantum search.

Findings

Relaxed the conditions on diffusion operators for quantum search.

Enabled a more general class of operators to be used in fast quantum search.

Improved the flexibility and potential efficiency of quantum search algorithms.

Abstract

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.