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An arbitrary two qubit gate can be constructed using local operations with CNOT gate. Are there other ways to implement these gates in this manner? In particular, can I decompose a two qubit gate in terms of local operations and SWAP gate? (The problem is solved if we can just write CNOT gate in terms of SWAP and local operations).

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As the other answer mentioned, the two-qubit gate that you use must create some entanglement, which the SWAP gate never does. So the SWAP cannot be used in this way.

However, any gate that creates entanglement can be used. See this paper for a method of doing so.

(What makes a gate one that creates entanglement? Assume it is a unitary $U$. If there exists a product state $|\psi\rangle|\phi\rangle$ such that $U|\psi\rangle|\phi\rangle$ is not a product state, then $U$ is entangling.)

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The swap gate doesn't entangle the two qubits, so any two-qubit unitary that can be implemented with swaps and local operations can also be implemented only with local operations. You can visualize this. Think of a swap gate surrounded by 4 local operations. You can move the swap gate to the right (or left) and permute the displaced local operations on the circuit. You can then remove the swap gate from the circuit by permitting the whole thing. If this circuit is a building block in other circuits, you must be a bit more careful, but the general idea still holds.

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