Abstract
Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on at mosta bits? Let 0≤a≤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 exactlya bits. We also prove a lower bound of n/2–1 for error-free Q * quantum protocols. Our results are obtained by employing basic tools from combinatorics and calculus to lower-bound the ranks of the appropriate matrices.
The research of the first author was supported in part by IQC University Professorship and CIAR, that of the second author in part by NSF grant CCR-01-05413, and that of the third author in part by NSF grant CCR-0208005 and NSF ITR Award CNS-0426683.
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Ambainis, A., Gasarch, W., Srinivasan, A., Utis, A. (2006). Lower Bounds on the Deterministic and Quantum Communication Complexities of Hamming-Distance Problems. In: Asano, T. (eds) Algorithms and Computation. ISAAC 2006. Lecture Notes in Computer Science, vol 4288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11940128_63
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DOI: https://doi.org/10.1007/11940128_63
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