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 $\varepsilon\in(0,1)$, there are binary permutation-invariant partial functions with quantum communication complexity $O_\varepsilon(\log\log n)$ and randomized communication complexity $\Omega_\varepsilon((\log n)^{1-\varepsilon})$. 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 $\Omega_\varepsilon(n^{1-\varepsilon})$. For graph-invariant partial functions on $v$-vertex graphs, we obtain $O_\varepsilon(\log v)$ quantum versus $\Omega_\varepsilon(v^{2-\varepsilon})$ randomized communication complexity. All separation protocols require no prior entanglement.