# Nested Grover’s Algorithm for Tree Search

## TL;DR

This paper introduces a nested Grover algorithm to improve quantum tree search efficiency for artificial intelligence applications.

## Contribution

A novel nested Grover algorithm is proposed to optimize quantum tree search by using partial candidate solutions and a concatenated oracle.

## Key findings

- The nested Grover algorithm reduces search costs from O(2m/2) to O(m·2m/4) for tree search with a branching factor of 2.
- The use of partial candidate solutions allows decomposition of quantum tree search using Grover’s algorithm.
- The approach provides a foundation for quantum artificial intelligence by avoiding incompatible heuristic functions.

## Abstract

We investigate optimizing quantum tree search algorithms by employing a nested Grover Algorithm. This approach seeks to enhance results compared to previous Grover-based methods by expanding the tree of partial assignments to a specific depth and conducting a quantum search within the subset of remaining assignments. The study explores the implications and constraints of this approach, providing a foundation for quantum artificial intelligence applications. Instead of utilizing conventional heuristic functions that are incompatible with quantum tree search, we introduce the partial candidate solution, which indicates a node at a specific depth of the tree. By employing such a function, we define the concatenated oracle, which enables us to decompose the quantum tree search using Grover’s algorithm. With a branching factor of 2 and a depth of m, the costs of Grover’s algorithm are O(2m/2). The concatenated oracle allows us to reduce the cost to O(m·2m/4) for m partial candidate solutions.