AI 中文总结
研究无噪声量子多对一网络上线性组合计算问题，提出的协议让发送方可访问更多量子位并预编码输入符号，支持更一般线性变换计算，通信成本至多为先前最佳结果，还证明了通信成本的次可加性。
AI 中文摘要
在这项工作中，我们考虑在无噪声量子多对一网络上计算线性组合的问题。有\(k\)个发送方Alice\(_1\)，…，Alice\(_k\)和一个接收方Bob。每个Alice\(_i\)有一个数据向量\(W_i \in \mathbb{F}^{m_i}\)，Bob想计算线性组合\(Y = V_1 W_1 + V_2 W_2 + \cdots + V_k W_k \in \mathbb{F}^m\)。发送方通过无噪声多对一量子网络向Bob传输量子态且彼此不通信，他们共享纠缠，Bob不共享。Allaix等人（2025）提出的N-Sum Box协议在一定约束下考虑了此问题。我们提出的协议通过让发送方访问更多量子位并明智地预编码输入符号，支持计算更一般的线性变换类。我们方案的通信成本至多为此领域先前最佳结果，在某些情况下严格更低。最后，我们证明通信成本在实例间是次可加的，具体识别出两个线性函数，单独计算它们的总成本严格大于联合计算它们的成本。
英文摘要
In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are $k$ senders, Alice$_1$, $\ldots$, Alice$_k$, and a single receiver, Bob. Each Alice$_i$ has a data vector $W_i \in \mathbb{F}^{m_i}$, where $\mathbb{F}$ is a finite field. Bob wants to compute the linear combination $Y = V_1 W_1 + V_2 W_2 + \cdots + V_k W_k \in \mathbb{F}^m$, where $V_i$ is an $m \times m_i$ matrix over $\mathbb{F}$. The senders transmit quantum states to Bob through a noiseless many-to-one quantum network, but they are not allowed to communicate with each other. The senders share entanglement among themselves, while Bob does not share this entanglement. They encode their classical information $W_i$, $i=1,\ldots,k$, into their local subsystems and transmit them to Bob so that he can recover $Y$ through a quantum measurement and subsequent post-processing. The N-Sum Box protocol proposed by Allaix et al. (2025) considers this problem under certain constraints on the linear combination and the distribution of the data vectors among the senders. We present protocols that support the computation of a more general class of linear transformations by giving the senders access to more qudits and allowing them to judiciously precode their input symbols. The communication cost of our schemes is at most that of the best-known prior results in this area and is strictly lower in certain cases. Finally, we demonstrate that the communication cost is subadditive across instances. Specifically, we identify two linear functions for which the total cost of computing them individually is strictly larger than the cost of computing them jointly.
