Quantum network coding for quantum repeaters

This paper considers quantum network coding, which is a recent technique that enables quantum information to be sent on complex networks at higher rates than straightforward routing strategies. Kobayashi et al. have recently showed the potential of this technique by demonstrating how any classical network coding protocol gives rise to a quantum network coding protocol. They nevertheless primarily focused on an abstract model, in which quantum resources such as additional quantum registers can be freely introduced at each node. In this work, we present a protocol for quantum network coding under weaker (and more practical) assumptions: Our new protocol works even for quantum networks where adjacent nodes initially share one Einstein-Podolsky-Rosen pair but cannot add any additional quantum registers or send any quantum information. A typical example of networks satisfying this assumption is quantum repeater networks, which are promising candidates for the implementation of large-scale quantum networks. Our results thus show that quantum network coding techniques can increase the transmission rate in such quantum networks as well.

Figure 1

The butterfly network and a classical coding protocol. Source nodes ($s_1$ and $s_2$) have for input bits $X$ and $Y$. The task is to send simultaneously bit $X$ from $s_1$ to $t_1$ and bit $Y$ from $s_2$ to $t_2$. This task is implemented by using a xor operation at relay node $r_1$ and target nodes ($t_1$ and $t_2$). Note this task cannot be solved by using routing.

Figure 2

An example of quantum repeater network. There are three quantum repeaters ($s$, $r$, and $t$), two free classical channels, and two EPR pairs $|{\Psi }^{+}{\rangle }_{AB}$ and $|{\Psi }^{+}{\rangle }_{CD}$ between $s$-$r$ and $r$-$t$, respectively.

Figure 3

The setting for our protocol. After the execution of the protocol, $s_1$ and $t_1$ (similarly, $s_2$ and $t_2$) share one EPR pair and are then able to perform quantum teleportation.

Figure 4

The quantum circuit for getting GHZ states from two EPR pairs by connection.

Figure 5

The circuit for connection:fanout.

Figure 6

The circuit for connection:add.

Figure 7

The circuit for removal.

Figure 8

The circuit for removal:add.

Figure 9

The correspondence between classical and quantum coding.

Figure 10

Overall view of the encoding circuit. Parenthetic numbers refer to the state after each step of the encoding procedure of Table .