Suppose during a smooth lattice merge operation, I perform one round of error correction, treating the two code patches as one single lattice that stretches across the intermediate qubits. Then, one of the weight-3 Z stabilizers at the boundary of the join I can measure $d-1$ times more. Directly after this, I perform the additional $d - 1$ rounds of syndrome measurements needed except without this one stabilizer. How does this impact the fault tolerance of my $d$ rounds of error correction? Do I still get sufficient volume for MWPM? If not, how much worse is it?
1 Answer
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If you run this circuit, you will find that the error rate of the logical measurement is not improving with code distance. It only takes one stabilizer being consistently wrong to get the logical measurement wrong. By doing fewer measurements of one stabilizer, you are making it trivial for it to be consistently wrong.
Pre-measuring the one stabilizer many times, without measuring the others, is not sufficient to fix this. Because, without the other stabilizers as cross-checks, two data errors is enough to make all the measurements of the one stabilizer incorrect.
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$\begingroup$ Can one use techniques adapted to defects in lattices to preserve the fault-tolerance of this? For instance, superplaquettes/superunits (arxiv.org/pdf/1607.00627.pdf)? This way, there is a particular physical qubit that is not necessarily involved in any stabilizer measurements beyond those rounds, as I desire. $\endgroup$– usermmCommented Apr 17 at 0:57
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