The generation and verification of quantum states are fundamental tasks for quantum information processing that have recently been investigated by Irani, Natarajan, Nirkhe, Rao and Yuen [CCC 2022], Rosenthal and Yuen [ITCS 2022], Metger and Yuen [FOCS 2023] under the term \emph{state synthesis}. This paper studies this concept from the viewpoint of quantum distributed computing, and especially distributed quantum Merlin-Arthur (dQMA) protocols. We first introduce a novel task, on a line, called state generation with distributed inputs (SGDI). In this task, the goal is to generate the quantum state $U\ket{\psi}$ at the rightmost node of the line, where $\ket{\psi}$ is a quantum state given at the leftmost node and $U$ is a unitary matrix whose description is distributed over the nodes of the line. We give a dQMA protocol for SGDI and utilize this protocol to construct a dQMA protocol for the Set Equality problem studied by Naor, Parter and Yogev [SODA 2020], and complement our protocol by showing classical lower bounds for this problem. Our second contribution is a dQMA protocol, based on a recent work by Zhu and Hayashi [Physical Review A, 2019], to create EPR-pairs between adjacent nodes of a network without quantum communication. As an application of this dQMA protocol, we prove a general result showing how to convert any dQMA protocol on an arbitrary network into another dQMA protocol where the verification stage does not require any quantum communication.

翻译：量子态的生成与验证是量子信息处理中的基础任务，近年来由Irani、Natarajan、Nirkhe、Rao与Yuen [CCC 2022]、Rosenthal与Yuen [ITCS 2022]、Metger与Yuen [FOCS 2023] 在“态合成”这一术语下进行了研究。本文从量子分布式计算的角度，特别是分布式量子梅林-亚瑟（dQMA）协议的角度，探讨了这一概念。我们首先在线性网络上引入一项新任务，称为“分布式输入态生成”（SGDI）。该任务的目标是在线性网络最右侧节点生成量子态 $U\ket{\psi}$，其中 $\ket{\psi}$ 是给定于最左侧节点的量子态，而 $U$ 是一个酉矩阵，其描述分布于线性网络的各节点。我们给出了针对SGDI的dQMA协议，并利用该协议为Naor、Parter与Yogev [SODA 2020] 研究的集合相等性问题构建了dQMA协议，同时通过展示该问题的经典下界对协议进行了补充。我们的第二个贡献是基于Zhu与Hayashi [Physical Review A, 2019] 近期工作提出的dQMA协议，该协议可在无需量子通信的情况下在网络相邻节点间创建EPR对。作为该dQMA协议的一个应用，我们证明了一个通用结论：可将任意网络上的任何dQMA协议转化为另一个验证阶段无需量子通信的dQMA协议。