maximising symmetry in systems of connectivity

[alan2here]

About the author

I'm Alan, I'm up to MSC in computer science, I've not got high ranking Math qualifications but I've always greatly enjoyed the subject.

Quick Summary

(Offer a brief pitch for the lesson you'd like to create. Who is the target audience, and what are you hoping they'll get from the lesson?)

Unlabelled graphs (also called networks) are nice simple ways of specifying a pattern of connectivity between individually indistinguishable things, for example "10 things connected in a cycle".

Graphs can have disconnected regions, but then they look like separate graphs and describe multiple different systems of things, so lets cover graphs that are "connected".

Uniform polyhedra are interesting because they're (sort of) maximally symmetric, no corner is special, you can never communicate where you are on them other than to point because every corner looks the same.

Regular graphs exist but they can easily be very messy and non-symmetric, the equivalent in graph theory to the property that uniform polyhedra have seems to be vertex transitivity. These graphs include the skeletons of all the uniform polyhedra, but also many interesting grids that loop back on themselves, the skeletons of all the "abstract polytopes", lots of discrete configuration spaces (uniform discrete systems?), and products of these under various different kinds of multiplication. This is a bit limited at small sizes and when the number of modes is prime, such as with 11 nodes, then it seems that all such graphs are circulant. However 12 is highly divisible and suddenly there are enough nodes for a vast plethora of possibilities, the well of possibilities here runs incredibly deep.

Exploring discrete configuration spaces (small ones without too much connectivity) have also yielded some quite spooky results. Many of the ones I've found look like uniform polyhedra with a couple of the faces missing, with a few small trees stuck on and a few huge cycles. It reminds me of group theory, with all the nice infinite families of prime groups, and then the super-bizarre sporadic groups, it also reminds me of the surreals, with lots of nice, tidy, uniform order, and then a family of strange values such as whole families of transfinites and the "it's zero but it's not equal to 0" values that often get random goofy names, and come across as if they are put together at random.

Finding these unlabelled, connected, vertex transitive graphs also relates to questions about what types of multiplication may be possible between graphs, overlapping cycles in permutations, and other areas.

Target medium

Whatever people want to make. I'm more for communicating my findings with someone so hopefully something can get made.

Contact details

alan2here@gmail.com Various other messengers are available, I can post them if you contact me directly.

[kamiloze2004]

How make Quantum Code within Classical Computer?

[alan2here]

Tell me about how this relates to quantum compute?

[leios]

Sorry for the spam! I have been trying to go through spam, but left this up because... Well, it could be related right?

I forgot all the details, but I thought there was a different between a quantum computer (what people have in industry) and a universal quantum computer. A universal quantum computer is one where all the qubits are interacting with each other. A non-universal quantum computer is a any case where that's not true (ie nearest neighbor interactions).

A study on how the connectivity graph between qubits relates to different algorithms could be exciting. My understanding is a universal quantum computer should have much less error correction leading to way better algorithmic implementations in practice?

This is only vaguely related to your proposal, but it might still be relevant and serve as some sort of physical application. Also: I didn't do quantum information during my PhD, I just simulated quantum matter. I might be wrong about the distinction between industrial quantum computers and universal ones.

[alan2here]

so for the non-universal sort, you must still want a consistent pattern of connectivity :) making this the available pallet

[alan2here]

so it must be fairly significant?