Quantum Algorithm for k-distinctness with Prior Knowledge on the Input

TL;DR

This paper presents a quantum algorithm for the k-distinctness problem that leverages prior input knowledge to achieve query complexity better than the best known unconditional algorithms, using the learning graph technique.

Contribution

It introduces a novel quantum algorithm for k-distinctness with prior input knowledge, improving query complexity over existing methods.

Findings

Runs in o(n^{3/4}) queries for fixed k with prior knowledge

Uses learning graph technique from arXiv:1105.4024

Achieves better complexity than the unconditional O(n^{k/(k+1)}) algorithm

Abstract

It is known that the dual of the general adversary bound can be used to build quantum query algorithms with optimal complexity. Despite this result, not many quantum algorithms have been designed this way. This paper shows another example of such algorithm. We use the learning graph technique from arXiv:1105.4024 to give a quantum algorithm for $k$-distinctness problem that runs in $o(n^{3/4})$ queries, for a fixed $k$, given some prior knowledge on the structure of the input. The best known quantum algorithm for the unconditional problem uses $O(n^{k/(k+1)})$ queries.