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Using lattice surgery technique, CNOT operation can be implemented as follows.

  • step 1. smooth merging between the control qubit and intermediate qubit
  • step 2. smooth splitting the merged qubit above
  • step 3. rough merging between the intermediate qubit and target qubit.

Steps 2 and 3 can be performed at the same time. Therefore, we can implement CNOT in $2d$ ($d$ is a distance) QEC cycles($d$ for smooth merging and $d$ for smooth splitting and rough merging at the same time).

I understood the bold sentence as follows: whether smooth operation on step 2 and merging operation on step 3 is performed first, the CNOT can be performed.

But when we assume rough merging is performed first and smooth splitting after then, the result is $a_1(a_2|00\rangle+b_2|11\rangle)+b_1(a_2|11\rangle+b_2|00\rangle)$ which is not CNOT operation. Note that control qubit state is $a_1|0\rangle+b_1|0\rangle$ and target qubit state is $a_2|0\rangle+b_2|0\rangle$.

So I think we should first perform smooth splitting and after $d$ cycle error correction, and then perform rough merging.

How CNOT can be performed in $2d$ error correction cycle and how can merging and splitting be performed at the same time?

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This is a lattice surgery CNOT:

enter image description here

Time goes from top to bottom. The cubes have size $d \times d \times d$ ($d$ data qubits by $d$ data qubits by $d$ rounds). Empty space is the normal surface code checkerboard pattern of crosschecks. Black surfaces are Z type boundaries in the surface code. White surfaces are X type boundaries in the surface code.

The top of the horizontal white bar is the smooth merge. The bottom of the horizontal white bar is the smooth split. The top of the horizontal black bar is the rough merge.

The key part of this diagram is that there is a black 3-way junction and a white 3-way junction, and that all surfaces of the same color that are not connected are a minimum distance apart. Any diagram with that pair of 3-way junctions and a minimum pipe size is be a protected surface code CNOT.

how can merging and splitting be performed at the same time?

The reason the split and the merge can happen simultaneously is because the vertical separator in the middle can be shrunk down to nothing without making anything too close to anything else:

enter image description here

How CNOT can be performed in $2d$ error correction cycle

Actually it can be done in $d$! In the above diagram, there are two unit-sized time steps required to finish the CNOT. But actually only one unit time step is needed. You can achieve this by moving one of the qubits past the other as part of doing the CNOT:

enter image description here

As you can maybe tell, it can get quite tricky to describe these constructions as a series of time steps. Like, what happens in the above diagram is that the qubits have opposite orientations and one of them suddenly grows past the other one while merging into it, then later it shrinks to a new location while splitting. Very weird. I find it's far better to just think of them as 3d objects, ignoring that one of the directions is time, when trying to make optimal constructions.

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  • $\begingroup$ Thank you for answering. I understood most of your explanation but some question remains. Is it that the last figure means the control qubit is moved to the other location in real qubit architecture or is it just for representing that the smooth splitting is performed first and merging later? (I have understood it can actually be done in 1 unit time step from your explanation). $\endgroup$ Aug 13 '21 at 9:05
  • $\begingroup$ @JonghyunLEE the physical qubits are not moving. The logical control qubit is being moved to different physical qubits. $\endgroup$ Aug 13 '21 at 14:22

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