Quantum communication and complexity

Hartmut Klauck

This paper surveys the field of quantum communication complexity. Some interesting recent results are collected concerning relations to classical communication, lower bound methods, one-way communication, and applications of quantum communication complexity.

Gilles Brassard A

Ran Raz

Communication complexity has become a central completity model.In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems.We show that for certain communication complezity problems quantum communication protocols are exponentially faster than classical ores.More explicitly, we give an example for a communication complexity relation (0~ promise problem) P such that: 1) The quantum communication complexity of P is O(log m). 2) The classical probabilistic communication complexity of P is Q(m'l'/ logm).(where m is the length of the inputs).This gives an ezponential gap between quantum communication complexity and classical probabilistic communication complexity.Only a quadratic gap was previously known.Our problem P is of geometrical nature, and is a finite precision variation of the following problem: Player I gets as input a unit vector z E R" and two orthogonal subspaces MO, MI c R". Player II gets as input an orthogonal matrixT : R" + R".Their goal is to answer 0 if T(x) E MO and 1 if T(x) E M,, (and any an~luer in any other case).We give an almost tight analysis for the quantum communication complexity and for the classical-probabilistic communication complexity of this problem.

Ronald de Wolf

Yunqi Huang, Zekun Ye

We study how symmetry affects quantum advantage in two-party communication complexity. For any partial function $f$ on length-$n$ strings over a fixed alphabet of size $q$, invariant under simultaneous coordinate permutations, we prove $R^{\mathrm{pub}}(f)=O_q(t\log(2+n/t))$, where $t=\min\{n,Q^{\ast}(f)^2\}$, $R^{\mathrm{pub}}$ denotes public-coin randomized communication complexity, and $Q^{\ast}$ denotes entanglement-assisted quantum communication complexity. This bound is tight for set disjointness and extends the binary-alphabet result of Guan et al. with a smaller logarithmic overhead. The exponent of the logarithmic factor cannot be reduced by any fixed positive amount: for every fixed $\varepsilon\in(0,1)$, there are binary permutation-invariant partial functions with quantum communication complexity $O_\varepsilon(\log\log n)$ and randomized communication complexity $Ω_\varepsilon((\log n)^{1-\varepsilon})$. We also characterize quantum communication complexity by a combinatorial parameter up to a logarithmic factor. Growing alphabets and graph symmetries admit exponential separations. For every fixed $\varepsilon\in(0,1)$, we exhibit permutation-invariant partial functions on length-$n$ strings over an $n$-symbol alphabet with quantum communication complexity $O_\varepsilon(\log n)$ and randomized communication complexity $Ω_\varepsilon(n^{1-\varepsilon})$. For graph-invariant partial functions on $v$-vertex graphs, we obtain $O_\varepsilon(\log v)$ quantum versus $Ω_\varepsilon(v^{2-\varepsilon})$ randomized communication complexity. All separation protocols require no prior entanglement.