发表机构
Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area; College of Physics and Optoelectronic Engineering, Shenzhen University; College of Computer and Data Science, Fuzhou University（粤港澳大湾区量子科学中心; 深圳大学物理与光电工程学院; 福州大学计算机与数据科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究对称性对双方通信复杂度中量子优势的约束，证明置换不变偏函数的量子复杂度与随机化复杂度关系，并构造指数级分离实例，推广了已有结果。
AI 中文摘要
我们研究对称性如何约束双方通信复杂度中的量子优势。对于任意固定大小为 $q$ 的字母表上、在同时坐标置换下不变的偏函数，我们证明公开硬币随机化与纠缠辅助量子通信复杂度满足 $R^{\mathrm{pub}}(f)=O_q(Q^{\ast}(f)^2\log n)$，其中 $n$ 为输入长度。我们还用一个组合参数在对数因子内刻画了量子通信复杂度。这些结果推广了 Guan 等人的二元字母表结果，并改进了其对数开销。输入长度依赖是必要的：对于每个固定的 $\varepsilon\in(0,1)$，二元置换不变偏函数可以具有量子复杂度 $O_\varepsilon(\log\log n)$ 和随机化复杂度 $\Omega_\varepsilon((\log n)^{1-\varepsilon})$。增长的字母表和图的对称性允许指数级分离。对于每个固定的 $\varepsilon\in(0,1)$，我们在长度为 $n$、字母表大小为 $n$ 的字符串上构造置换不变偏函数，其量子复杂度为 $O_\varepsilon(\log n)$，随机化复杂度为 $\Omega_\varepsilon(n^{1-\varepsilon})$。我们还在 $v$ 个顶点的图上构造图不变偏函数，其量子复杂度为 $O_\varepsilon(\log v)$，随机化复杂度为 $\Omega_\varepsilon(v^{2-\varepsilon})$。所有分离协议既不使用先验纠缠，也不使用共享随机性。
英文摘要
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.
