In "stabilizers measurement" stile of QECC, one can increase the code distance by creating additional (unentangled) qubits, and subsequently measure the stabilizers of the former code along with the stabilizers which shares the old qubits and the new ones. This is mainly done after state injection, where the "injection" part is performed on a small distance code whereas the final code should be of high-distance to further maintain small fault-tolerant logical error rate.
My question is: what about MBQC? For example, assuming I have a RHG cube with distance d1. Now I did some injection method (see e.g. here) and I want to increase the code distance to d2. If I understand correctly, the way to do it is with lattice surgery: this means you first merge this d1 code to a larger (d2-d1) code in one direction (say, enlarging the X logical) and then in the other direction. However, during all of this process, including the O(d) additional layers in time required for error-correction, your fault distance is actually d1. I think this is also true for "stabilizers measurement" stile, but there you can increase the entire code in one step + d stabilizer rounds (e.g. here), and not in two steps.
Is my description correct? And if so, is there another, more efficient method?
EDIT:
In RHG I meant the Raussendorf-Harrington-Goyal cube, i.e. just the standard topological cluster state which is the simple MBQC equivalent of the surface code (this). Also, I focused in MBQC in the context of graph state - no stabilizer measurements, just a bunch of qubits initialized in the |+> state and some CZ gates between them, along with a single-qubit X-basis measurements.
In those cases, I was wondering if there is an alternative to lattice surgery when one wants to change the code distance from d1 to d2 (specifically I was interested in the stage after a magic-state injection).
