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While I'm studying about Qubit allocation (mapping), I wonder what the differences between SWAP and Bridge gates are. I know that a SWAP gate and Bridge gate can be represented with 3 CNOT gates and 4 CNOT gates respectively. Changing the qubit location of the hardware topology using SWAP is visually visible. However, Bridge gate case, I couldn't.

Here is my Questions

  1. What is the difference between SWAP gate and Bridge gate
  2. Is there any gates that can change physical qubit location of hardware topology except SWAP and Bridge gates?
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Regarding to your first question:

As you said, the SWAP gate can be decomposed into 3 CNOTs as:

enter image description here

and the Bridge gate can be decomposed into 4 CNOTs as:

enter image description here

From the circuits above, we can see that the SWAP gate only involve 2 qubits. It Swap them. Whereas the Bridge gate is to perform a CNOT gate between two non-adjacent qubits with one qubit between them.

But also note that:

enter image description here

This is important because suppose your device have the following architecture:

enter image description here

which doesn't have qubit-0, $q_0$, connected to qubit 2, $q_2$, but you need to apply a CNOT gate between them in the middle of the circuit you are executing. Then, instead of naively using the SWAP gate to swap qubit 1,$q_1$, with qubit 2, $q_2$, then do a CNOT on $q_0$ and $q_1$, then apply another SWAP gate to swap qubit 1 and qubit 2 back to their original spot, we can just use the Bridge gate.

And you can combine both gate together too. For instance, you want to execute a CNOT between qubit 0, and qubit 4. For the above hardware architecture, you can SWAP qubit 3 and qubit 4, perform Bridge gate, then SWAP qubit 3 and qubit 4 back again. That is,

enter image description here

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  • $\begingroup$ First, thank you for your answer. I have some questions about your answer. So Bridge make a new path between non-adjacent qubits? If so, is the path reusable or temporary one? And 'Whereas the Bridge gate is to perform a CNOT gate between two non-adjacent qubits with one qubit between them.' this sentence, can there be only one qubit between two non-adjacent qubits? How about two qubits? $\endgroup$
    – 김동민
    Feb 24, 2021 at 4:28
  • $\begingroup$ The Bridge gate is intended to allow you to do the CNOT gate operation when two qubits are not connected directly.... but SWAP gate actually swaps the qubit and once you swap them, you can do whatever you want, then swap back. But if your intend is to only do a CNOT gate between two non-connected qubit then the Bridge gate is better. Now, if the qubits are far apart, like qubit 0 and qubit 4 in the above architecture, you can't apply the bridge gate directly either... you have to do some extra steps. $\endgroup$
    – KAJ226
    Feb 24, 2021 at 4:33
  • $\begingroup$ So, you mean we can use Bridge if there is only one qubit between two non-adjacent qubits? And also, since Bridge doesn't change qubit location, When I want to do only CNOT gate, then Bridge is better than SWAP. $\endgroup$
    – 김동민
    Feb 24, 2021 at 4:54
  • $\begingroup$ Yes, that is how I see it. $\endgroup$
    – KAJ226
    Feb 24, 2021 at 4:59
  • $\begingroup$ So, is there any quantum gates or method to do 2-qubit gates without changing hardware-topology(Architecture)? SWAP changed it but Bridge didn't. $\endgroup$
    – 김동민
    Feb 24, 2021 at 5:01

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