布尔隐藏匹配问题推广的量子优势刻画

中文摘要 AI 辅助

我们研究$f$-布尔隐藏划分问题的单向通信复杂度。其中，Alice获得一个$n$比特字符串，Bob获得其索引的$\Omega(n)$个不相交块以及一组标签。在承诺这些块上$f$的求值要么与所有标签一致，要么与所有标签不一致的前提下，Bob必须利用Alice发送的一条消息来确定属于哪种情况。该问题推广了布尔隐藏匹配和隐藏超匹配问题，后者对应于$f$为奇偶函数的特殊情况。我们依据$f$的符号度$d$建立了$f$-布尔隐藏划分问题的经典和量子通信界，证明了Doriguello和Montanaro（TQC 2020）的一个猜想。他们先前的工作在$d \le 1$时给出了经典协议的对数通信上界，在$d \le 2$时给出了量子协议的对数通信上界，并对具有更大符号度的某些结构化函数给出了多项式下界。我们证明，对于每个符号度为$d \ge 2$的$f$，该问题的随机经典通信复杂度为$\Theta(n^{1-1/d})$，而其量子通信复杂度介于$\Omega(n^{1-2/d})$和$\bO{n^{1-1/\lceil d/2 \rceil} \log n}$之间。这完全刻画了允许多项式量子优势的函数$f$，即那些满足$d \ge 2$的函数，并由$d = 2$的情形给出了一个新的无限指数分离函数族。

英文摘要

We study the one-way communication complexity of the $f$-Boolean Hidden Partition problem. Here, Alice is given an $n$-bit string and Bob is given $Ω(n)$ disjoint blocks of its indices together with a string of labels. Under the promise that the evaluations of $f$ on these blocks either agree with all of the labels or disagree with all of them, Bob must determine which is the case using a single message from Alice. This problem generalizes the Boolean Hidden Matching and Hidden Hypermatching problems, which capture the special case where $f$ is the parity function. We establish classical and quantum communication bounds for $f$-Boolean Hidden Partition in terms of the sign degree $d$ of $f$, proving a conjecture of Doriguello and Montanaro (TQC 2020). Their prior work gave logarithmic communication upper bounds for classical protocols when $d \le 1$ and quantum protocols when $d \le 2$, as well as polynomial lower bounds for certain structured functions with larger sign degree. We show that for every $f$ of sign degree $d \ge 2$, the randomized classical communication complexity of this problem is $Θ(n^{1-1/d})$ while its quantum communication complexity lies between $Ω(n^{1-2/d})$ and $\bO{n^{1-1/\lceil d/2 \rceil} \log n}$. This completely characterizes the functions $f$ admitting polynomial quantum advantage as those for which $d \ge 2$, with a new infinite family of exponential separations given by the case $d = 2$.

发表机构

• Boston University（波士顿大学）
• HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利研究网络阿尔弗雷德·雷尼数学研究所）
• National University of Singapore（新加坡国立大学）
• IRIF (CNRS & Université Paris Cité)（IRIF（法国国家科学研究中心与巴黎西岱大学））

机构由 AI 辅助整理，请以论文原文为准。