# What is the applicability of quantum network coding?

As the title suggests, I want to know what the applicability of quantum network coding is, besides the EPR pair construction between distant pairs of 'Users-Targets'.

Can quantum network coding be used for computation?

• Do you have a link to a paper describing what quantum network coding is? Commented Jun 13, 2018 at 3:53
• @CraigGidney I can't find a pedagogic paper but here you have: link link Commented Jun 13, 2018 at 4:09
• Those papers are talking about quantum networks, but they don't define "quantum network coding". Are you just asking what a quantum internet could be useful for? Commented Jun 13, 2018 at 4:25
• @CraigGidney I haven't read it, but this looks relevant. Commented Jun 13, 2018 at 5:02
• @Blue Oh, I see, it refers to complex protocols for sending data to multiple parties under network constraints that force you to do stuff like xor the messages together over some links in order to maximize throughput. Commented Jun 13, 2018 at 5:45

Network coding — both classical network coding, and quantum network coding — is an approach to distributing information by performing simple operations at nodes in a network, acting on input signals and transmitting the outputs to other nodes. To put it another way, network coding is an approach to distributing information using a communications network if we treat it as a logical circuit, albeit where the 'gates' performed at each node may be a bit more powerful than just AND, OR, CNOT, or the like.

In principle, we can use the setting of network coding to perform non-trivial computations by an appropriate choice of operations (gates) at the nodes. Network coding does not usually allow the freedom to also choose the structure of the network itself (i.e. the circuit topology), as this is usually given as an input parameter to a given network coding problem. But there will still be some range of computations which a given network can admit, not all of which will serve merely to distribute information.

In the particular case of quantum network coding, the detail that things are to be done in a distributed (and presumably coherent) manner does add wrinkles to how you can manage to accomplish things. However, if we allow classical communication between nodes in the network as well — either allowing classical messages to move both forward and backward within the coding network or in an all-to-all manner — then you can perform coherent quantum network coding for the k-pairs problem [1] or an arbitrary network coding problem [2] respectively, provided that a classical network protocol exists for the same problem in the same network: and furthermore, the way this is done can be seen to essentially be Measurement Based Quantum Computation (MBQC), as Martin Roeteller and I showed [3]. Conversely, it is fairly clear that for any MBQC procedure, there is a corresponding coding network topology which allows that procedure to be realised.

It follows that, while the details are a bit pickier than in the classical case, quantum network coding can be viewed as a setting in which to do universal computation, specifically via MBQC, at least so long as auxiliary classical communication is allowed (with somewhat fewer constraints than on quantum communication).

[1] Constructing Quantum Network Coding Schemes from Classical Nonlinear Protocols. Kobayashi et al. (2010). [arXiv:1012.4583]

[2] General Scheme for Perfect Quantum Network Coding with Free Classical Communication. Kobayashi et al. (2009). [arXiv:0908.1457]

[3] Quantum linear network coding as one-way quantum computation. de Beaudrap & Roetteler (2014). [arXiv:1403.3533]