Learning-Graph-Based Quantum Algorithm for k-distinctness

TL;DR

This paper introduces a new quantum algorithm for the k-distinctness problem that improves query complexity using a modified learning graph approach, and also presents an efficient algorithm for the graph collision problem.

Contribution

It develops a novel quantum algorithm for k-distinctness with improved query complexity and simplifies the analysis without needing prior input information.

Findings

Achieves $O(n^{1-2^{k-2}/(2^k-1)})$ query complexity for k-distinctness.

Introduces an $O(\sqrt{n}\alpha^{1/6})$ algorithm for graph collision.

Simplifies the complexity analysis compared to previous methods.

Abstract

We present a quantum algorithm solving the $k$-distinctness problem in $O(n^{1-2^{k-2}/(2^k-1)})$ queries with a bounded error. This improves the previous $O(n^{k/(k+1)})$-query algorithm by Ambainis. The construction uses a modified learning graph approach. Compared to the recent paper by Belovs and Lee arXiv:1108.3022, the algorithm doesn't require any prior information on the input, and the complexity analysis is much simpler. Additionally, we introduce an $O(\sqrt{n}\alpha^{1/6})$ algorithm for the graph collision problem where $\alpha$ is the independence number of the graph.