Brainstorming ideas from my former life as a theoretical physicist (once a physicists, always a physicist?) that might serve as some sort of design inspiration.
Quantum error correcting codes
Peter Shor came up with the original 9 qubit quantum error correcting code
By trying to put it on a lattice, I discovered that it could be changed into a different kind of code, that was both more complicated conceptually, but simpler when implemented
And when you try the same thing on Archimedian tilings, you get all sorts of fun codes
Communication cost of simulating entanglement
Protocols for simulating measurements on a Bell pair use some interesting geometry on spheres
To figure out how these protocols work, Ben Toner and I used some fun integrals
Back in my Ph.D. days I thought a lot about exchange interactions. This is an interaction that acts to exchange two qubits, and while it would seem to not be that powerful it turns out that you can build a quantum computer just using exchange interactions. These looks like hills, but its really how to perform a controlled-phase gate using only exchange interactions
Into the Third Dimension
The two dimensional subsystem code, has a few three dimensional cousins, of which here is one.
Local subsystem codes (unpublished)
This is a dictionary of sorts for turning a quantum circuit into a quantum subsystem code
It yields interesting constructions. Here a cat state is used to measure a four qubit stabilizer