Faulty quantum computing with diagrams

Quantum computing is already strange. And measurement-based quantum computing is even weirder, at least to me. Remember how the very act of looking at something quantum changes it? Well, you can essentially use this to your advantage and do computation in this way.

First, you set up a massive quantum state, with all the underlying qubits entangled in some way. And then, you just selectively observe certain qubits, and this collapses them in a way that carves out a quantum circuit in the underlying state. So instead of having to carefully prepare a sophisticated arrangement of quantum bits, you prepare a much less sophisticated one and then “look” at it cleverly to get the same thing.

One way to show that this really does give you the same thing, is by adopting a diagrammatic language. So instead of doing lots of linear algebra and writing everywhere, you make simple drawings. There is a genuine way to represent the algebraic mess with simple dots and lines (they call them spiders). This language is the ZX-Calculus, and it is an exceptionally simple way to genuinely reason about quantum computing.

An example of how diagrams of spiders translate to quantum states.

It gets even more fun, when you realise that the graphical diagrams, at least in the measurement-based setting, map pretty much directly to the physical representation of qubits. So the way people arrange these qubits on a lattice, can be drawn out in a similar way with these mathematical diagrams. This is really useful, because in practice when you lay out these qubits in the real world, you can get little faults and missing parts. And these can be copied over to the mathematical diagram as well.

Percolation theory, which studies the behaviour of graphs when you remove vertices or edges, helps describe properties of these errors. And interestingly, there are sudden limitations in computational power when you get too many of these errors.

The physical percolation of the lattice has a direct translation to the ZX-Calculus.

This work was completed as part of my Master’s degree in Theoretical Physics at Oxford under the supervision of Prof. Renaud Lambiotte and Prof. Matthew Hoban.