Computation-Aided Classical-Quantum Multiple Access to Boost Network Communication Speeds

A multiple access channel (MAC) consists of multiple senders simultaneously transmitting their messages to a single receiver. For the classical-quantum case (CQ MAC), achievable rates are known assuming that all the messages are decoded, a common assumption in quantum network design. However, such a conventional design approach ignores the global network structure, i.e., the network topology. When a CQ MAC is given as a part of quantum network communication, this work shows that computation properties can be used to boost communication speeds with code design dependent on the network topology. We quantify achievable quantum communication rates of codes with the computation property for a two-sender CQ MAC. When the two-sender CQ MAC is a boson coherent channel with binary discrete modulation, we show that it achieves the maximum possible communication rate (the single-user capacity), which cannot be achieved with conventional design. Further, such a rate can be achieved by different detection methods: quantum (with and without quantum memory), on-off photon counting, and homodyne (each at different photon power). Finally, we describe two practical applications, one of which cryptographic.

Figure 1

Physical example of quantum one-hop relay network model. In this example, the relay is a satellite. The two senders $A$ and $B$ are located in distant places so that they cannot directly communicate to each other. In phase 1, the two senders access simultaneously the satellite. Hence, the uplink is a CQ MAC (the network bottleneck) that has two senders $A$ and $B$ and a single decoding system on the satellite. In phase 2, the relay communicates to each sender via two point-to-point CQ channels towards senders $A$ and $B$. In this work we show that conventional decoding is not optimal (i.e., cannot achieve the capacity of the network) and introduce computation (COMP) codes that allow the actual sum of (quantized) electromagnetic fields in the CQ MAC of phase 1 to be decoded, enabling network capacity to be achieved.

Figure 2

Illustration of the operation of COMP codes proposed in this work, compared to conventional SD codes when two senders simultaneously transmitting their messages over a CQ MAC. The diagram on the left illustrates the concept of COMP codes. The main objective of our work is the design of the required COMP codes when only the modulo sum $M_A\oplus M_B$ needs to be reliably transmitted over the CQ MAC. For comparison, the conventional approach [13, 26] is shown on the right: the receiving node decodes both messages, $M_A$ and $M_B$ using conventional simultaneous-decodable codes, denoted here as SD codes.

Figure 3

Left: achievable communication rates with collective measurement, with SD codes, Eq. (17) (in blue) and with COMP codes for different measurements: collective as Eq. (19) (in red), on-off as Eq. (20) (in dashed red) and homodyne as Eq. (26) (in dashed point red). Right: the corresponding gains of COMP codes with different measurements over SD codes (up to $33.33\mathrm{\%}$). As a useful reference, the single-sender Holevo quantity [i.e. $h(e^{-|\alpha {|}^2}\cosh |\alpha {|}^2)$] gives the upper bound.

Figure 4

Left: schematic illustration of the quantum compute-and-forward relaying strategy, whereby Bob and Alice exchange their messages over a CQ MAC channel with COMP codes. Right: schematic illustration of our method for symmetric private information retrieval (SPIR) with two servers based on CQ MAC with COMP codes. Observe the very different role of the CQ MAC in the two applications while they employ the same network model, the quantum one-hop relay network model.

Figure 5

Another example where COMP codes offer an advantage, the butterfly network. Node $v_6$ ($v_5$) intends to receive the message from node $v_1$ ($v_2$). In the optimal method, nodes $v_3$, $v_5$, and $v_6$ recover only the modulo sum of the two information messages. A red arrow shows a part of CQ MAC, in which our COMP code can be applied.