Generalized Grover's algorithm for multiple phase inversion states

TL;DR

This paper generalizes Grover's algorithm to handle multiple target states by deriving an exact solution using eigenspectrum analysis and quantum phase estimation, achieving near-optimal search complexity.

Contribution

It introduces a new framework for Grover's algorithm with multiple phase-inverted states, providing an exact solution and an alternative approach via quantum phase estimation.

Findings

Exact solution for multiple target states in Grover's algorithm

Derived initial state for generalized Grover evolution

Achieves near-optimal search complexity of √(D/M^α)

Abstract

Grover's algorithm is a quantum search algorithm that proceeds by repeated applications of the Grover operator and the Oracle until the state evolves to one of the target states. In the standard version of the algorithm, the Grover operator inverts the sign on only one state. Here we provide an exact solution to the problem of performing Grover's search where the Grover operator inverts the sign on N states. We show the underlying structure in terms of the eigenspectrum of the generalized Hamiltonian, and derive an appropriate initial state to perform the Grover evolution. This allows us to use the quantum phase estimation algorithm to solve the search problem in this generalized case, completely bypassing the Grover algorithm altogether. We obtain a time complexity of this case of $\sqrt(D/M^\alpha)$ where D is the search space dimension, M is the number of target states, and $\alpha$ ~ 1, which is close to the optimal scaling.