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I am trying to understand the lattice surgery technique performed in the paper Entangling Logical Qubits with Lattice Surgery and I'm running into some confusion with the merging procedure. The setup is as follows:

  1. First, two logical qubits are separately encoded in 2x2 surface codes (Figure A below). The stabilizer generators are defined as Pauli X's on the vertices of the orange plaquettes and Pauli Z's on the vertices of aqua plaquettes. For example lattice A has stabilizer group generated by $\langle Z_1 Z_2, Z_3Z_4, X_1X_2X_3X_4 \rangle $

  2. Next, the two lattices are merged by projecting onto a joint eigenstate of $X_L^AX_L^B$. This is accomplished by measuring the 'merging stabilizers' $X_3X_5$ and $X_4X_6$, which are just the new X type stabilizers that would have been included in the joint lattice.

The paper claims that the merged code is now stabilized by $\langle Z_1 Z_2, X_1X_2X_3X_4, Z_3Z_4Z_5Z_6, X_5X_6X_7X_8, X_3X_5, X_4X_6, Z_7Z_8 \rangle$. This would require that the encoded qubit is in a $+1$ eigenstate of both merging stabilizers (by the fact that it should be stabilized by those operators). However, my understanding is that by measuring those merging stabilizers, we only guarantee that the merged qubit is in an eigenstate of the joint measurement, meaning it could be in a -1 eigenstate. This would suggest to me that the code is not really stabilized by those operators. In fact, the paper acknowledges this fact, but still claims the code is stabilized by the typical stabilizers you would write down based on the color of the plaquettes. Any idea what's going on here? Is it typical to use the merging stabilizer measurements to correct the encoded state to be in the +1 eigenspace?

Any thoughts are appreciated, thanks in advance!

lattice surgery on 2x2 lattices

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As far as computation goes, it doesn't matter whether the state stabilized by a stabilizer or its negation. What matters is that you know the sign of the stabilizer with confidence, via the measurements you are performing and the error correction process.

The situation is basically analogous to teleportation, where the transmission has a 50% chance of bit flipping the qubit and also a 50% chance of phase flipping the qubit. As long as you know whether or not the bit flip and phase flip occurred, you can account for them and continue onward.

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