Tight Quantum Lower Bound for k-Distinctness

TL;DR

This paper presents a new quantum query lower bound framework that generalizes existing methods and applies it to establish a tight lower bound for the k-Distinctness problem.

Contribution

It introduces a novel framework inspired by the compressed oracle technique that subsumes the polynomial method and applies it to k-Distinctness.

Findings

Established the first tight quantum query lower bound for k-Distinctness.

Framework does not rely on oracles and uses Fourier basis to analyze knowledge.

Allows arbitrary input distributions, broadening applicability.

Abstract

In this paper, we introduce a new quantum query lower bound framework. It is inspired by Zhandry's compressed oracle technique, but it also subsumes the polynomial method as a special case. Compared to Zhandry's technique, our approach has two key differences. First, we do not use any oracles (except for the standard input oracle), and define ``knowledge'' directly through the expansion of the state of the algorithm in the Fourier basis. Second, we allow arbitrary probability distributions of inputs. We show how this framework behaves on the problem of finding equal elements in the input string. In particular, we demonstrate its power by proving a first tight quantum query lower bound for the k-Distinctness problem.