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How do you do a fault tolerant Bell measurement in the surface code?

Experimentally how do you do a Bell state measurement? Is it just a CNOT and a Hadamard and then a measurement in the computational basis? Or is it even simpler than that because it is easy to measure in any basis of stabilizer states that you want?

Relatedly, how do you do a Bell state measurement on encoded qubits fault tolerantly? Do you need a transversal CNOT and transversal Hadamard and then just measure in the computational basis?

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In the surface code, when using lattice surgery, a Bell basis measurement is performed via a lattice surgery pair measurement in the X basis and then transversal single qubit Z basis measurements. (It also works if you do Z pair measurement followed by transversal X measurement.) In other words, it's decomposed into:

MXX 0 1
MZ 0
MZ 1

In a spacetime defect diagram of the lattice surgery, this just looks like the two qubits' pipes joining together:

enter image description here

The crossbar is the MXX and the incoming pipes terminating with a red ending (instead of the pipes continuing after the crossbar) is the two MZ operations.

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  • $\begingroup$ hmm I see so is the idea that to build a fault tolerant Bell measurement you need some fault tolerant $ CNOT $ on the surface code and one sort of fault tolerant $ CNOT $ is the transversal $ CNOT $ but here you are instead using the lattice surgery $ CNOT $ from arxiv.org/pdf/1111.4022 to do fault tolerant $ CNOT $ and so this is a sort of lattice surgery based fault tolerant Bell measurement? $\endgroup$
    – Ian Gershon Teixeira
    Commented Jul 25 at 15:35
  • $\begingroup$ @IanGershonTeixeira you have it the wrong way around. In the surface code parity measurements come before controlled not. Logical Cnot is built out of parity measurements not the other way around. So there are no controlled not gates here, only measurements. $\endgroup$
    – Craig Gidney
    Commented Jul 25 at 15:52
  • $\begingroup$ ok thanks for the response, clearly I need to read up more on fault tolerance for surface codes. Do you know if any of these techniques for fault tolerant Bell measurement can be adapted to other CSS codes or simple stabilizer codes like the $ [[5,1,3]]$? $\endgroup$
    – Ian Gershon Teixeira
    Commented Jul 25 at 16:24

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