Non-unitary extension of Grover's search algorithm

TL;DR

This paper introduces a non-unitary extension of Grover's algorithm that achieves quadratic speedup using a bigger rotation, leveraging quantum singular value transform and additional resources.

Contribution

It presents a novel non-unitary approach to Grover's search that reaches the Grover bound with minimal extra resources, expanding quantum search techniques.

Findings

Achieves $O( oot N)$ complexity with quantum singular value transform.

Requires only one extra qubit to reach Grover's bound.

Classical repetition is still needed for post-selection success.

Abstract

We have developed a non-unitary extension of Grover's search algorithm by changing the hidden geometry of Hilbert space carried by diffusion operator. Our algorithm finds the solution for search problem by performing a unique bigger rotation rather than small rotations in order polynomial times in the size $N$ of search space. We analyze the complexity of implementing the non-unitary operation and we observed that the price paid by performing this rotation is due the normalization. In Kraus operator approach we need $O(N)$ repetition of the algorithm to have a chance of measuring a solution in a post-selection, this is no better than the classical solution. However, the quantum singular value transform in addition with block encoding and Chebyshev polynomial approximation, we got complexity $O(\sqrt{N})$ and reach the Grover's bound with an extra resource of one single qubit, compared with the standard Grover's algorithm.