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I was curious to whether the two logical qubits on the toric code can be entangled through, for instance, a logical CNOT operation. However, I cannot find any information on this, only how you can do this on the surface code through braiding defects. I was wondering if people already looked into this because I haven't been able to find anything yet in literature. Does lattice surgery provide an option?

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  • $\begingroup$ This isn't the conventional way to perform gates in these systems, but two copies of the surface code admit a transversal CNOT gate; applying CNOT gates between physical qubit applies a logical CNOT gate. This probably works for two copies of the toric code as well. $\endgroup$
    – user34722
    Mar 26, 2023 at 23:31

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I cannot find any information on this [...] Does lattice surgery provide an option?

If you go to google scholar and search "lattice surgery" it brings up the relevant papers.

The very first paper on lattice surgery, "Surface code quantum computing by lattice surgery", explained how to perform the CNOT. It is decomposed into parity measurements and the parity measurements are performed by merging and splitting patches. That said, this paper is kinda hard to understand.

The CNOT is also covered in "Low overhead quantum computation using lattice surgery" and in "A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery", which are the papers you want to read if you want to understand lattice surgery.

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  • $\begingroup$ Does this work for the two logical qubits on the toric code? I think this assumes you have two copies of the surface code, not one toric code. It's not obvious to me what to do to entangle the two logical qubits of the toric code. $\endgroup$ Mar 10, 2023 at 21:37
  • $\begingroup$ Jahan Claes is right, I couldn't find information on the toric code. But the surface code suffices for now, I did indeed find that first paper but it was really technical. I will have to dig deep into it and take some time to figure it out. I'll also take a look at the other papers, thank you very much! $\endgroup$
    – JoJo
    Mar 10, 2023 at 21:39

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