Generalized Grover Search Algorithm for Arbitrary Initial Amplitude Distribution

TL;DR

This paper generalizes Grover's quantum search algorithm to work with any initial amplitude distribution, deriving exact solutions and optimal measurement times based on initial state statistics.

Contribution

It introduces a generalized framework for Grover's algorithm accommodating arbitrary initial amplitudes, with exact amplitude evolution equations and measurement time optimization.

Findings

Derived exact solutions for amplitude evolution.

Established optimal measurement time T rom initial amplitudes.

Bound the measurement probability based on initial amplitude standard deviation.

Abstract

Grover's algorithm for quantum searching of a database is generalized to deal with arbitrary initial amplitude distributions. First order linear difference equations are found for the time evolution of the amplitudes of the r marked and N-r unmarked states. These equations are solved exactly. An expression for the optimal measurement time T \sim O(\sqrt{N/r}) is derived which is shown to depend only on the initial average amplitudes of the marked and unmarked states. A bound on the probability of measuring a marked state is derived, which depends only on the standard deviation of the initial amplitude distributions of the marked or unmarked states.