Faster quantum walk algorithm for the two dimensional spatial search

TL;DR

This paper introduces a faster quantum walk algorithm for 2D spatial search, reducing the complexity from O(√N log N) to O(√N log N) steps by controlling the walk with an ancilla qubit.

Contribution

The paper presents a novel quantum walk algorithm that improves search efficiency on a 2D lattice using ancilla qubit control.

Findings

Achieves O(√N log N) search steps, improving over previous algorithms.

Demonstrates the effectiveness of ancilla qubit control in quantum walks.

Abstract

We consider the problem of finding a desired item out of $N$ items arranged on the sites of a two-dimensional lattice of size $\sqrt{N} \times \sqrt{N}$. The previous quantum walk based algorithms take $O(\sqrt{N}\log N)$ steps to solve this problem, and it is an open question whether the performance can be improved. We present a new algorithm which solves the problem in $O(\sqrt{N\log N})$ steps, thus giving an $O(\sqrt{\log N})$ improvement over the known algorithms. The improvement is achieved by controlling the quantum walk on the lattice using an ancilla qubit.