A Time-Efficient Quantum Walk for 3-Distinctness Using Nested Updates

TL;DR

This paper introduces a novel quantum walk algorithm with nested updates for 3-Distinctness, achieving optimal query complexity and improved time efficiency over previous methods.

Contribution

It extends the quantum walk framework to include nested updates and applies this to develop a more time-efficient algorithm for 3-Distinctness.

Findings

Query complexity ~O(n^{5/7}) matching the best known bounds

Time complexity ~O(n^{5/7}) improving over previous ~O(n^{3/4})

Introduces nested updates to quantum walk framework

Abstract

We present an extension to the quantum walk search framework that facilitates quantum walks with nested updates. We apply it to give a quantum walk algorithm for 3-Distinctness with query complexity ~O(n^{5/7}), matching the best known upper bound (obtained via learning graphs) up to log factors. Furthermore, our algorithm has time complexity ~O(n^{5/7}), improving the previous ~O(n^{3/4}).