# A Block-Sensitivity Lower Bound for Quantum Testing Hamming Distance

## TL;DR

This paper establishes a quantum query complexity lower bound of a( d7 n/g) for the Gap-Hamming distance problem, advancing understanding of quantum limits in string distance testing.

## Contribution

It introduces a novel lower bound for quantum testing of Hamming distance using block sensitivity and reduction techniques.

## Key findings

- Quantum lower bound of a( d7 n/g) for Gap-Hamming distance
- Uses combinatorial block sensitivity and threshold function reduction
- Provides insights into quantum query complexity limits

## Abstract

The Gap-Hamming distance problem is the promise problem of deciding if the Hamming distance $h$ between two strings of length $n$ is greater than $a$ or less than $b$, where the gap $g=|a-b|\geq 1$ and $a$ and $b$ could depend on $n$. In this short note, we give a lower bound of $\Omega( \sqrt{n/g})$ on the quantum query complexity of computing the Gap-Hamming distance between two given strings of lenght $n$. The proof is a combinatorial argument based on block sensitivity and a reduction from a threshold function.