Quantum Advantage of Permutation-Invariant Functions in Communication Complexity

Abstract

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 $\epsilon \in (0,1)$, there are binary permutation-invariant partial functions with quantum communication complexity $O_{\epsilon }(\log \log n)$ and randomized communication complexity ${\Omega }_{\epsilon }((\log n{)}^{1-\epsilon })$. 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 $\epsilon \in (0,1)$, we exhibit permutation-invariant partial functions on length-$n$ strings over an $n$-symbol alphabet with quantum communication complexity $O_{\epsilon }(\log n)$ and randomized communication complexity ${\Omega }_{\epsilon }(n^{1-\epsilon })$. For graph-invariant partial functions on $v$-vertex graphs, we obtain $O_{\epsilon }(\log v)$ quantum versus ${\Omega }_{\epsilon }(v^{2-\epsilon })$ randomized communication complexity. All separation protocols require no prior entanglement.