Lower bounds on the Deterministic and Quantum Communication Complexity of Hamming Distance

Andris Ambainis; William Gasarch; Aravind Srinavasan; Andrey Utis

Home/Archive/arXiv

arXiv · cs/0411076

Lower bounds on the Deterministic and Quantum Communication Complexity of Hamming Distance

Andris Ambainis·William Gasarch·Aravind Srinavasan·Andrey Utis

View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

cs.CC quant-ph

Keep this discovery

BibTeX RIS

Andris Ambainis, William Gasarch, Aravind Srinavasan, Andrey Utis. 2004-11-20. Lower bounds on the Deterministic and Quantum Communication Complexity of Hamming Distance. https://arxiv.org/abs/cs/0411076

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗

arXiv2026

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

cs.CC↗

arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗