# Additive decomposition of iterative quantum search operations in the Grover-type algorithm

## TL;DR

This paper introduces an additive decomposition method for iterative Grover-type quantum search operations, expressing the process as a linear combination of simpler operators to reduce computational complexity.

## Contribution

It presents a novel additive decomposition scheme for iterative quantum search, enabling more efficient implementation and a shortcut gate for the same outcome.

## Key findings

- Decomposition expresses k iterations as a sum of single-operator terms.
- Final state can be represented with a single oracle and phase-rotation.
- Proposes a shortcut gate to reduce computational load.

## Abstract

In the Grover-type quantum search process a search operator is iteratively applied, say, k times, on the initial database state. We present an additive decomposition scheme such that the iteration process is expressed, in the computational space, as a linear combination of k operators, each of which consists of a single Grover-search followed by an overall phase-rotation. The value of k and the rotation phase are the same as those determined in the framework of the search with certainty. We further show that the final state can be expressed in terms of a single oracle operator of the Grover-search and phase-rotation factors. We discuss how the additive form can be utilized so that it effectively reduces the computational load of the iterative search, and we propose an effective shortcut gate that realizes the same outcome as the iterative search.