Lower bounds on the Deterministic and Quantum Communication Complexity of Hamming Distance
<|reference_start|>Lower bounds on the Deterministic and Quantum Communication Complexity of Hamming Distance: Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on \textit{at most} a bits? Let 0\leq a\leq n-1. We show that any deterministic protocol, as well as any error-free quantum protocol (C* version), for this problem requires at least n-2 bits of communication. We show the same bounds for the problem of determining if two strings differ in exactly a bits. We also prove a lower bound of n/2-1 for error-free Q* quantum protocols. Our results are obtained by lower-bounding the ranks of the appropriate matrices.<|reference_end|>
@article{ambainis2004lower,
    title={Lower bounds on the Deterministic and Quantum Communication Complexity
  of Hamming Distance},
    author={Andris Ambainis, William Gasarch, Aravind Srinavasan, Andrey Utis},
    journal={arXiv preprint arXiv:cs/0411076},
    year={2004},
    archivePrefix={arXiv},
    eprint={cs/0411076},
    primaryClass={cs.CC quant-ph}
}