What is the point of defect diagrams?

Question

Gidney's Understanding Defect Diagrams has been helpful to better understand defect diagrams. The pictures are cool, I'm just struggling to connect them to specific questions we can answer with them.

My key question is: what types of questions can you answer with a defect diagram? What is the workflow?

Q1: Is the following correct about the use cases and workflow of defect diagrams?

Use cases

1. Validating operations -- as in here, operations like lattice surgery preserve some spacetime stabilizers. Thus, demonstrating that these spacetime stabilizers remain constant via defect diagrams is analogous to a proof of validity.
2. Analyzing error strings in spacetime -- as in Figure 8 here, defect diagrams can be used to identify spacetime errors which cause logical errors / classify these errors.

Defect diagram workflow

1. For each point in time, draw the defect diagram of the surface code. These will likely be deformed over time, in which case boundaries will change as time progresses.
2. Through the computation, correlation surfaces can be identified. These surfaces represent stabilizers in spacetime, i.e. the product of the stabilizers at multiple times which should be constant.
3. Correlation surfaces, like boundaries, will absorb some excitations but not others. Pauli errors of a different type than the surface will cause an incorrect measurement.

Q2: As seen in Figure 15, what types of questions can be answered / what is the workflow for this defect diagram?

Answer 1

Q1: Is the following correct about the use cases and workflow of defect diagrams?

Yes, those look correct.

I would add "visualizing cost" as a core use case. The geometry of defect diagrams specifies an amount of space and time needed to implement a computation. If I were to instead give you a ZX graph or circuit of the computation, embedding it into spacetime might add unexpected overhead when attempting to link up all the pipes and junctions correctly. With a defect diagram all that overhead is already included.

Q2: As seen in Figure 15, what types of questions can be answered / what is the workflow for this defect diagram?

This figure is intended to be an overview of the various simulations that were done in the paper. It's not really supposed to surprise you with new answers, but rather to present an interesting perspective on what was done. When I look at it I see examples of step-by-step defect diagrams creating different shapes in spacetime, and a demonstration of how variations that look innocuous in the moment (like mirroring) can have important topological implications in spacetime (like linking rings, creating a logical error mechanism).