In a seminal paper AGP, the authors proved the threshold theorem and obtained a threshold for concatenated Steane code $[[7^k,1,3^k]]$. One thing to note is that, to ensure fault-tolerance, error correction gadgets are inserted in between logical gates and some exRecs are constructed. I've been looking for similar thing for the surface code, but failed to find analogous discussion in literature. So my question is, for a circuit encoded in the surface code, do I also perform error-correction in between logical gate implementations to ensure fault-tolerance? Or how should I do it otherwise?
i.e. what's the procedure for simulating an ideal circuit with surface code, is it also
FT prep of a big codepatch -- EC -- FT logical gate -- EC -- FT logical gate?
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Doing operations with surface code lattice surgery doesn't look like gate-then-correct-then-gate-then-correct. It looks like transitioning between different surface code configurations, pausing at each one to measure the stabilizers enough times become confident in products of areas of stabilizers. The logical gates end up embedded into the spacetime topology, so that locally everything just looks like normal planar surface code stabilizer measurement.
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$\begingroup$ Yes I understand lattice surgery. However, it seems to me LS is performing checks/or measurements only on boundary and ancillary rows of qubits, what if the qubits on the codepatch decohere? For the AGP scheme, if in the middle of computation faults occur on data qubits, it can also be corrected via intermediate EC. In surface code case, if not EC-gate-EC I'm not sure how LS can make sure it's fault-tolerant. In other words, I think if you only use LS, you need the data qubits to have very long coherence time? Or am I understanding something wrong? $\endgroup$ Commented Jun 3 at 15:54
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1$\begingroup$ @AndyLiuin You don't just measure things on the boundaries, you're always measuring all the stabilizers. And there's no need to correct errors just-in-time with lattice surgery; you only need to figure them out after the fact. You just need to do the measurements so that they can be figured out later. $\endgroup$ Commented Jun 3 at 16:03
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$\begingroup$ Having looked back the LS surgery paper, around equations (14)(15) they perform d rounds of error correction after each merging and splitting. So I suppose this is also in some sense Gate -- EC ? $\endgroup$ Commented Jun 3 at 18:19
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$\begingroup$ @AndyLiuin Yes, that would be a reasonable interpretation. $\endgroup$ Commented Jun 3 at 20:15