Imagine you have merged two surface codes with lattice surgery by measuring all the stabilizers and treating both surface codes as one surface code, then are we doing error correction before or after applying CNOT gate or not? If yes, are we doing error correction on each surface code separately or together after the lattice surgery?
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are we doing error correction before or after applying CNOT gate
The error correction is not time sensitive so this doesn't matter. You can do it after. You can even do it a hundred surface code cycles later. You can even do it after the lattice surgery is long done. Pauli corrections can be arbitrarily backdated in stabilizer circuits, like surface code lattice surgery.
There's no deadline. The only requirement is that in the limit you are solving the decoding problem at the same rate it's being produced. Throughout is the hard constraint; latency is a soft constraint.
are we doing error correction on each surface code separately or together
Together. You treat the stitched surface codes like one big long surface code.
There are performance reasons to want to solve the two halves mostly-seperately, but those methods always involve a final step that carefully combines the two pieces to behave as if they had always been one piece.
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$\begingroup$ Thanks, Craig for your answer. I have one more question about the actual lattice surgery process. Do we need to entangle the borders of the two surface codes with the physical qubits that we use for lattice surgery? This paper (arxiv.org/pdf/2302.01296) talks about entangling them but the actual lattice surgery paper (iopscience.iop.org/article/10.1088/1367-2630/14/12/123011/pdf) does not mention anything about entangling them. So I am confused a little bit. $\endgroup$ Commented May 22 at 20:58
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$\begingroup$ @ShahroozPooryousef There's a lot of different ways to do it, and a lot of different ways to describe it, so I'm not sure how to answer something as vague as "does it entangle the qubits". $\endgroup$ Commented May 22 at 21:02
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$\begingroup$ thanks @Craig Gidney. I did not know there were different ways of doing lattice surgery. Sorry if my comment was not clear. Is there a link explaining these different ways? Can you point me to some links, please? I would appreciate it. $\endgroup$ Commented May 22 at 23:30
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$\begingroup$ @ShahroozPooryousef I don't really have links for it. It's just... for example, the two logical patches may be far apart or not, or they may have opposite boundary orientation or not, or may be on separate chips requiring communication or not. Lattice surgery just means you turn the two patches into one patch and then back somehow. $\endgroup$ Commented May 23 at 0:21