Faster quantum searching with almost arbitrary operators

TL;DR

This paper introduces a quantum search algorithm that allows the use of almost arbitrary diffusion operators, maintaining near-optimal query complexity by leveraging quantum Fourier transform, thus broadening practical applicability in quantum computing.

Contribution

It presents a method to use arbitrary diffusion operators in quantum search without losing optimal query complexity by employing quantum Fourier transform techniques.

Findings

Maintains near-optimal query complexity with arbitrary operators

Uses quantum Fourier transform to compensate for increased query count

Advantage when oracle is more expensive than diffusion operator

Abstract

Grover search algorithm drives a quantum system from an initial state to a desired final state by using selective phase inversions of these two states. In (1), we studied a generalization of Grover algorithm which relaxes the assumption of the efficient implementation of the selective phase inversion of the initial state, also known as diffusion operator. This assumption is known to become a serious handicap in cases of physical interest (2,3,4,5). Our general search algorithm works with almost arbitrary diffusion operator with only restriction of having the initial state as one of its eigenstates. The price that we pay for using arbitrary operator is an increase in the number of oracle queries by a factor of order of B, where B is a characteristic of the eigenspectrum of diffusion operator and it can be large in some situations. Here we show that by using quantum fourier transform, we can regain the optimal query complexity of Grover algorithm without losing the freedom of using arbitrary diffusion operators for quantum searching. However, the total number of operators required by algorithm is still order of B times more than that of Grover algorithm. So our algorithm offers advantage only if oracle operator is computationally more expensive than diffusion operator, which is true in most search problems.