A Block-Sensitivity Lower Bound for Quantum Testing Hamming Distance

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| ≥ 1 and a and b could depend on n. In this short note, we give a lower bound of Ω( √ n/g) on the quantum query complexity of computing the GapHamming 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.