How to generate matrix for swap(a, b) gate for n qubits

Question

I am trying to simulate a swap gate that swaps two qubits of indices a and b where there are n qubits total. I understand how to make a truth table and generate a matrix based off of that for each individual case but I'm having trouble implementing a general way to do this for an arbitrary total number of qubits. Any help is appreciated.

Answer 1

I am assuming you are trying to figure out a matrix that represents this type of circuit:

In such case, first realize the decomposition of SWAP gate:

Therefore, if you know how to implement the circuit below then you will pretty much done.

To implement the circuit above, recall the definition of $CNOT_{1,2}$ (The first qubit is the controlled and the second qubit is the target):

$$ CNOT_{1,2} = |0\rangle \langle 0| \otimes I + |1 \rangle \langle 1| \otimes X $$

But here we wanted to implement $CNOT_{2,4}$ instead since the second qubit is the controlled and the fourth qubit is the target. In such case, you can write it as

$$ CNOT_{2,4} = I \otimes |0\rangle \langle 0| \otimes I \otimes I + I \otimes |1 \rangle \langle 1| \otimes I \otimes X $$

Also note that by definition:

$$ CNOT_{2,1} = I \otimes |0\rangle \langle 0 | + X \otimes |1\rangle \langle 1 |$$

I think you can finish it off now.