I have been thinking a bit on what consensus on a quantum network means and whether the solution for von Mises-Fisher consensus has any good application towards quantum information theory. There are 4 ways to define Consensus on a quantum network (Mazzarella, et al. 2015):

- Sigma Expectation Consensus -Same distribution of measurements at each node -"Same expectation”
- Reduced State Consensus -When all nodes have the same density matrix or state -“Same Marginal Distributions”
- Symmetric State Consensus -When all nodes stay the same after conjugation by unitary permutation matrix -“Joint Probability distributions invariant to permutation”
- Single Sigma-Measurement Consensus -Similar to Sigma-Expectation, but results must be the same for each trial/measurement. -“Perfect agreement for each measurement”

I will be looking at how these networks communicate to reach consensus. This seems to be a major difference between classical networks and quantum networks, as in quantum networks, we cannot measure the information directly, lest we collapse the superposition. Also, it is not possible to communicate quantum information through a classical channel. I have also seen a few things about the stabilizer/Heisenberg picture of quantum computing, from (Gottesman, 1998). Maybe this would be useful, but would it force us to use the matrix von Mises-Fisher distribution? If so, then we would need to formulate an equation and/or algorithm for the consensus by KL Average for matrix von Mises-Fisher distributions first.

A few questions that I need to consider:

- How can I represent the qubit with the von Mises-Fisher distribution such that I can carry the uncertainty information and the expected state of the qubit such that the network can evolve according to a slightly modified Lindblad master equation (Shi, et al. 2015) or a type of gossip algorithm (Mazarella, et al. 2015)?
- If I cannot do (1), is there a communication structure that will conserve the quantumness of the network while also allowing consensus to be achieved?
- The von Mises-Fisher representation of a qubit will have the advantage in that it can characterize arbitrary bit/amplitude and phase noise in a qubit, unlike surface codes that only correct for bit and phase flips. However, surface codes (Fowler, et al. 2012) are the ones that are most popularly used (including Google's Sycamore quantum computer), and there appears to be a "digitization in errors".