Quantum Distributed Complexity of Set Disjointness on a Line
Frederic Magniez, Ashwin Nayak
备注：
24 pages, 2 figures. In v3, background on Theorem 3.5 (from prior work) was included in an appendix. In v4, a detail in the statement of Theorem 3.5 has been corrected, and corresponding changes have been made in the rest of the paper. The results remain unchanged. In v5, extended main theorems to the entanglement-assisted setting, added explanations and clarifications, corrected typos
发表信息：
ACM Transactions on Computation Theory, Volume 14, Issue 1, March 2022, Article No.: 5, pp 1-22
Set Disjointness on a Line is a variant of the Set Disjointness problem in a distributed computing scenario with $d+1$ processors arranged on a path of length $d$. It was introduced by Le Gall and Magniez (PODC 2018) for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model. However, they were only able to provide a lower bound when the local memory used by the processors on the intermediate vertices of the path consists of O$( \log n)$ qubits for $n$-bit inputs. We prove an unconditional lower bound of $\widetildeΩ\big(\sqrt[3]{n d^2}+\sqrt{n} \, \big)$ rounds for Set Disjointness on a Line with $d + 1$ processors. The result gives us a new lower bound of $\widetildeΩ \big( \sqrt[3]{nδ^2}+\sqrt{n} \, \big)$ on the number of rounds required for computing the diameter $δ$ of any $n$-node network with quantum messages of size O$(\log n)$ in the CONGEST model.
We draw a connection between the distributed computing scenario above and a new model of query complexity. The information-theoretic technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to the number of rounds in this query model. We provide an algorithm for Set Disjointness in this query model with round complexity that matches the round lower bound stated above, up to a polylogarithmic factor. This presents a barrier for obtaining a better round lower bound for Set Disjointness on the Line. At the same time, it hints at the possibility of better communication protocols for the problem.
我们证明下界的复杂性措施，如布尔函数的近似程度和布尔矩阵的近似秩，使用量子参数。我们证明了这些下界使用量子查询算法的组合组测试问题。我们证明了对于任何函数f，计算f的n个副本的OR的近似程度是Ω（sqrt{n}）乘以f的近似程度，这是最优的。在我们的工作之前，没有这样的一般结果，甚至AND函数的OR下限也是在2013年才解决的。然后，我们证明了一个类似的结果，在通信复杂性，显示的对数的近似秩（或更准确地说，近似gamma_2范数）的F：X x Y -> {0，1}增长的一个因素Ω ~（sqrt{n}）时，我们采取的或的n个副本的F，这也是本质上最优的。作为推论，我们给出了Razborov著名的Ω（sqrt{n}）下界的量子通信复杂性的不相交性问题的一个新的证明。最后，我们将这两个结果从与OR函数的组合推广到与任意对称函数的组合，在这种情况下也得到了接近最优的下界。
Quantum Advantage of Permutation-Invariant Functions in Communication Complexity
Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）；College of Physics and Optoelectronic Engineering, Shenzhen University（深圳大学物理与光电工程学院）；College of Computer and Data Science, Fuzhou University（福州大学计算机与数据科学学院）
Communication Complexity of Distributed Unitary Synthesis
State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；School of Computer Science and Technology, University of Chinese Academy of Sciences（中国科学院大学计算机科学与技术学院）
