Communication Efficiency of Summation Over a Two-Layer Quantum MAC

To compute a summation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(\Sigma)$</tex> of independent classical data streams from multiple servers over a quantum multiple access channel (QMAC), every server must directly send its quantum system containing multiple qudits to the user. Previous work <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Sigma$</tex> -QMAC in [1] considers that the servers are entangled. In this work, the maximum rate, defined as the maximum number of dits of the sum computed per qudit transmitted, is derived, and a quantum coding scheme is designed to achieve it. In practice, a server is not always directly connected to the user, necessitating a relay between them for connectivity. Motivated by this, we consider a two-layer quantum MAC (2QMAC) model, where a user is connected to a layer of mirrors, and one of the mirrors (acting as a relay) is connected to a layer of servers. Quantum systems within the same layer are entangled, whereas quantum systems across different layers are independent. In <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Sigma-2\text{QMAC}$</tex>, deriving the maximum rates of two layers simultaneously is challenging since the rates of the two layers affect each other. To address this, we first derive the maximum rate of the first layer by neglecting the second layer. Second, to relate the rates of the two layers, we derive the maximum rate of the second layer for a given feasible rate (less than the maximum rate) of the first layer. In addition, a joint coding scheme based on the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$N$</tex>-sum box abstraction is designed to achieve the maximum rate of the second layer.