# 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? – Craig Gidney Jun 13 '18 at 3:53
• @CraigGidney I can't find a pedagogic paper but here you have: link link – Ivanovitch Jun 13 '18 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? – Craig Gidney Jun 13 '18 at 4:25
• @CraigGidney I haven't read it, but this looks relevant. – Sanchayan Dutta Jun 13 '18 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. – Craig Gidney Jun 13 '18 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]

• Thank you for the answer, Niel. Just one thing: when linking papers, if time permits, just add the titles of the papers along with main authors and publication dates, so that later when paper use search engines to look up discussions related to those papers they will notice our site. That would also be would be helpful if someone is using the site's native search. Also, it seems one of the papers you referenced is co-authored by you. It would be nice if you add a "disclosure" regarding that, in the answer itself. :) – Sanchayan Dutta Jun 13 '18 at 13:25
• @Blue: as you ask nicely, I acknowledge my authorship. But you will also have to ask me again in the future. On occasion (ideally often!) someone on this site will know the answer to a question because they have had a hand in writing the relevant article, and I don't consider such meta-data relevant to the quality of an answer. An answer here is focused and of high-quality – or isn't – regardless of the relationship of a respondent with the author list of any references. Should it not be enough to refrain from abusing the forum, and to write good, focused answers to reasonable questions? – Niel de Beaudrap Jun 13 '18 at 17:19
• I really hope that you're not taking this personally. However, it's generally considered an unwritten rule all over SE, that "you should disclose personal connections whenever you reference something you're involved with". See this discussion on Physics SE for example. Again, I repeat, I very much appreciate your contributions to the site! Don't get me wrong. – Sanchayan Dutta Jun 13 '18 at 17:40
• @Blue: I don't really take it personally, and whenever I write something which is tentative, I certainly do flag my relationship to what I have written. I also think that I understand the motivation for asking people to identify their relationship with the cited work (i.e. making it easier to identify and discourage gratuitous self-promotion). I disagree with that policy, as I would prefer to simply discourage unfocused and off-topic content generally, without the personal element entering into the calculus. But if it is a well-established policy across SE, I will try to adhere to it. – Niel de Beaudrap Jun 15 '18 at 12:32