Quantum nondeterministic distributed computing was recently introduced as dQMA (distributed quantum Merlin-Arthur) protocols by Fraigniaud, Le Gall, Nishimura and Paz (ITCS 2021). In dQMA protocols, with the help of quantum proofs and local communication, nodes on a network verify a global property of the network. Fraigniaud et al. showed that, when the network size is small, there exists an exponential separation in proof size between distributed classical and quantum verification protocols, for the equality problem, where the verifiers check if all the data owned by a subset of them are identical. In this paper, we further investigate and characterize the power of the dQMA protocols for various decision problems. First, we give a more efficient dQMA protocol for the equality problem with a simpler analysis. This is done by adding a symmetrization step on each node and exploiting properties of the permutation test, which is a generalization of the SWAP test. We also show a quantum advantage for the equality problem on path networks still persists even when the network size is large, by considering ``relay points'' between extreme nodes. Second, we show that even in a general network, there exist efficient dQMA protocols for the ranking verification problem, the Hamming distance problem, and more problems that derive from efficient quantum one-way communication protocols. Third, in a line network, we construct an efficient dQMA protocol for a problem that has an efficient two-party QMA communication protocol. Finally, we obtain the first lower bounds on the proof and communication cost of dQMA protocols. To prove a lower bound on the equality problem, we show any dQMA protocol with an entangled proof between nodes can be simulated with a dQMA protocol with a separable proof between nodes by using a QMA communication-complete problem introduced by Raz and Shpilka (CCC 2004).

翻译：量子非确定性分布式计算最近由Fraigniaud、Le Gall、Nishimura和Paz（ITCS 2021）以dQMA（分布式量子Merlin-Arthur）协议的形式提出。在dQMA协议中，借助量子证明和局部通信，网络中的节点验证网络的全局性质。Fraigniaud等人指出，当网络规模较小时，对于相等性问题（即验证者检查其子集所拥有的所有数据是否相同），分布式经典验证协议与量子验证协议在证明大小上存在指数级分离。本文进一步研究并刻画了dQMA协议在各种判定问题上的能力。首先，我们针对相等性问题给出了一个更高效的dQMA协议，并附带更简洁的分析。这是通过在每个节点上添加对称化步骤并利用置换测试（SWAP测试的推广）的性质实现的。我们还证明了，通过考虑极端节点之间的“中继点”，即使在大型网络中，路径网络上的相等性问题仍保持量子优势。其次，我们表明，即使在通用网络中，对于排序验证问题、汉明距离问题以及更多源自高效量子单向通信协议的问题，也存在高效的dQMA协议。第三，在线性网络中，我们为一个拥有高效两方QMA通信协议的问题构造了高效的dQMA协议。最后，我们得到了dQMA协议在证明和通信开销上的首个下界。为证明相等性问题的下界，我们展示了任何节点间使用纠缠证明的dQMA协议，均可通过使用由Raz和Shpilka（CCC 2004）引入的QMA通信完全问题，模拟为节点间使用可分离证明的dQMA协议。