# Proofs that lattice surgery does what it claims to do

I'm interested in making a little list of papers that include a proof that lattice surgery really does what it claims to do; namely, performs a pairwise logical measurement. The original paper (http://arxiv.org/abs/1111.4022) gives a detailed sketch of what the protocol does but perhaps doesn't quite constitute a rigorous proof that it really works. A couple of papers (http://arxiv.org/abs/1810.10037, http://arxiv.org/abs/1704.08670) prove that the merge step of the protocol is a projector at the logical level, and the splitting step is its adjoint. A third uses low-level ZX-calculus (http://arxiv.org/abs/2204.14038), and a fourth uses the stabilizer formalism (http://arxiv.org/abs/2006.03071).

Are there other papers out there you know of that contain a proof that lattice surgery works as described? Only needs to be for a specific example code (e.g. surface code of a certain size). Thanks in advance!

The trick to proving surface code constructions are correct is to draw them as topological defect diagrams, find the correlation surfaces these diagrams support, and check that those surfaces propagate identically to the stabilizer flows of the desired operation. I don't know of a paper that explains this, but I've given a talk on it (video) (slides).

The benefit of this style of proof is that it's efficient to compute (scales to tens of thousands of logical qubits), it generalizes across styles of surface code computation (braiding vs lattice surgery vs others), and it's invariant under topological deformations and spacetime rotations.

For an $$XX$$ parity measurement, the relevant surfaces you must show exist are:

$$X_1 \rightarrow X_1$$

$$X_2 \rightarrow X_2$$

$$Z_1Z_2 \rightarrow Z_1Z_2$$

$$X_1 X_2 \rightarrow I$$