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I am reading 2013, Horsman D. et al. - Surface code quantum computing by lattice surgery. I have been trying to understand how the outcome of rough lattice merging, which is discussed using both stabilizer formalism and logical qubit formalism, can be retrieved by carrying out operations on physical qubits. My current understanding is that, while logically lattice merging is equivalent to measuring the product of logical operators $X_1X_2$, this measurement procedure would not involve the merge qubits introduced at the boundary at all, effectively leaving them in their initial $|0\rangle$ state.

I am therefore assuming that either:

a) one physically measures each $A_{V}$ separately and shows that, depending on the sign of the product of these $A_V$, you can identify the output state with the logical state of the extended code as shown in eq. (4).

b) one measures $X_1X_2$ (independent of the merge qubits) and that a subsequent decoding procedure is required to fix the potential syndrome along the line of merging qubits.

I know the stabilizer analysis in the appendix A is much clearer and is supporting the former hypothesis, but then why does the main text use the latter and, most importantly, why is it correct?

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this measurement procedure would not involve the merge qubits introduced at the boundary at all, effectively leaving them in their initial |0⟩ state.

You have to repeatedly measure the XXXX and ZZZZ stabilizers touching these qubits to get the measurement result while maintaining fault tolerance. The XXXX stabilizer measurements anticommute with the initial |0⟩ state and so do not leave the qubits in that state.

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  • $\begingroup$ Thanks for the clarification. What confuses me is the following: say we prepare the system in logical state |0⟩|0⟩ and do rough merging. This is equivalent to measuring the N XXXX stabilizers (as in the circuit in figure 2b of the paper). There are 2^N possible final states depending on which stabilizers is in eigenvalue +1 and which in -1. These are all different from each other, but the outcome of the merging should be either one of the two possible states |0⟩ + (-1)^{M} |1⟩ where M is the product of all the N measurement outcomes. Are there some steps I am missing in my calculations? $\endgroup$ Sep 6, 2023 at 13:23
  • $\begingroup$ @GiovanniCanossa No, that's right. The logical measurement is 1 distributed degree of freedom amongst many. That's part of what makes it fault tolerant. $\endgroup$ Sep 6, 2023 at 16:42
  • $\begingroup$ I still don't see how. The outcome of the measurements involved in the merging are 2^{N} : we have 2^{N-1} possible different states with M=+1 and the same amount with eigenvalue M=-1. If we take two outcomes with M=+1, these will trivially not be identical to each other because they have different stabilizer eigenvalues and therefore the two cannot both be the global state |0⟩ + |1⟩. That's why I originally thought that one instead measures the product of the XXXX stabilizers and not each stabilizer individually. $\endgroup$ Sep 7, 2023 at 11:26
  • $\begingroup$ @GiovanniCanossa It doesn't matter that there are multiple possible outputs, what matters is you can tell them apart and account for the differences. Each case is individually solvable, and you know which case you've ended up in. $\endgroup$ Sep 7, 2023 at 17:16

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