# A novel quantum grid search algorithm and its application

## TL;DR

This paper introduces a quantum grid search algorithm that leverages multiple Grover operators in parallel, providing exponential speedup for high-dimensional optimization problems compared to classical methods.

## Contribution

The paper presents a novel quantum grid search algorithm that extends Grover's search to solve complex, high-dimensional optimization problems with exponential efficiency gains.

## Key findings

- Offers exponential improvement over classical search algorithms.
- Uses parallel Grover operators for efficient search.
- Applicable to high-dimensional trajectory optimization.

## Abstract

In this paper we present a novel quantum algorithm, namely the quantum grid search algorithm, to solve a special search problem. Suppose $ k $ non-empty buckets are given, such that each bucket contains some marked and some unmarked items. In one trial an item is selected from each of the $ k $ buckets. If every selected item is a marked item, then the search is considered successful. This search problem can also be formulated as the problem of finding a "marked path" associated with specified bounds on a discrete grid. Our algorithm essentially uses several Grover search operators in parallel to efficiently solve such problems. We also present an extension of our algorithm combined with a binary search algorithm in order to efficiently solve global trajectory optimization problems. Estimates of the expected run times of the algorithms are also presented, and it is proved that our proposed algorithms offer exponential improvement over pure classical search algorithms, while a traditional Grover's search algorithm offers only a quadratic speedup. We note that this gain comes at the cost of increased complexity of the quantum circuitry. The implication of such exponential gains in performance is that many high dimensional optimization problems, which are intractable for classical computers, can be efficiently solved by our proposed quantum grid search algorithm.