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The Gap-Hamming distance problem is the promise problem of deciding if theHamming distance $h$ between two strings of length $n$ is greater than $a$ orless 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 thequantum query complexity of computing the Gap-Hamming distance between twogiven strings of lenght $n$. The proof is a combinatorial argument based onblock sensitivity and a reduction from a threshold function.

A Block-Sensitivity Lower Bound for Quantum Testing Hamming Distance