Precoding-based protocols for entanglement assisted linear computation over a quantum many-to-one network

In this work, we consider the problem of computing a linear combination over a noiseless quantum many-to-one network. There are k senders, Alice₁, 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 = V₁ W₁ + V₂ W₂ + 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.