We know that a SWAP gate needs three CNOT gates but I have seen papers which say that a SWAP gate can be achieved by necessary rewiring can are not counted towards the final quantum cost. I am confused about whether to count the SWAP gate (which accounts for the cost of 3 CNOT gates) or not to count its cost at all. This is necessary for the overall depth of the circuit and also gate count.
- In Implementing Grover oracles for quantum key search on AES and LowMC, section 4.2 ShiftRow and RotByte, they said "we consider rewiring as free and do not include it in our cost estimates."
- In Efficient Implementation of PRESENT and GIFT on Quantum Computers, section 3.1.3. Permutation of PRESENT Block Cipher, they said ". Otherwise, this can be implemented with qubits relabeling. Therefore, Swap gates are not counted as gate cost. Therefore, the quantum cost for Permutation of PRESENT block cipher is zero."