# Is the SWAP gate a Clifford Gate? How would I express it using the Clifford Gate generators?

By my calculations, it looks like the SWAP gate is a Clifford Gate. See the following table: I follow the same method as in this paper for showing a gate is a Clifford Gate. I got the above table by performing calculations in Qiskit. How would I express the SWAP gate in terms of the generators of the Clifford group? The generators are the $$H$$ gate, the $$S$$ gate and the $$CNOT$$ gate.

It's well known that you can make a swap out of three CNOTs. For reference, Stim's gate documentation includes H+S+CX decompositions of a lot of Clifford gates including the swap:

Stabilizer Generators:

X_ -> +_X
Z_ -> +_Z
_X -> +X_
_Z -> +Z_

Unitary Matrix:

[+1  ,     ,     ,     ]
[    ,     , +1  ,     ]
[    , +1  ,     ,     ]
[    ,     ,     , +1  ]

Decomposition (into H, S, CX, M, R):

# The following circuit is equivalent (up to global phase) to SWAP 0 1
CNOT 0 1
CNOT 1 0
CNOT 0 1

• thank you for the answer :) I saw before the the $C_{2}NOT_{1}$ and $C_{1}NOT_{2}$ gates can be used to make a $SWAP$ gate. My question then is: how is the $C_{2}NOT_{1}$ generated using the generators of the Clifford group? Nov 11 at 18:24
• $C_2NOT_1=(H\otimes H)\circ C_1NOT_2\circ(H\otimes H)$ Nov 11 at 18:26
• @QuantumGuy123 That gate is considered to be in the gate set. But if you really want to go out of your way you can just surround a CNOT with Hs instead of switching the target and control roles. Nov 11 at 18:26
• hey @CraigGidney, do you (or perhaps someone you know) know the answer to my other question on Cliffords?: quantumcomputing.stackexchange.com/questions/21993/… Nov 19 at 22:14

Yes, SWAP is a Clifford gate.

Your proof is correct. By definition, an $$n$$-qubit gate $$U$$ is Clifford if $$UPU^\dagger\in G_n$$ for all $$P\in G_n$$ where $$G_n$$ is the $$n$$-qubit Pauli group. However, it is easy to see that we only need to check that $$UQU^\dagger\in G_n$$ for $$Q$$ that are generators of $$G_n$$. Moreover, $$G_n$$ is generated by $$\{i, Z_k, X_k\,|\,k=1,\dots,n\}$$ where $$X_k$$ denotes the tensor product of Pauli $$X$$ on the $$k$$th qubit and identity applied to all other qubits and similarly for $$Z_k$$. Since $$UiU^\dagger=iI\in G_n$$ for all $$U$$, we only have to check that $$UX_kU^\dagger\in G_n$$ and $$UZ_kU^\dagger\in G_n$$ for $$k=1,\dots,n$$. This is exactly what the table in the question accomplishes.

It is easy to check that

$$\text{SWAP} = C_1NOT_2 \circ C_2NOT_1 \circ C_1NOT_2\tag1$$

where $$C_iNOT_j$$ denotes the CNOT gate with qubit $$i$$ as control and $$j$$ as the target, e.g. by applying both sides of $$(1)$$ to the computational basis states.

• thank you for the answer :) Nov 11 at 18:23
• see my comment in the other answer. Nov 11 at 18:25
• do you (or perhaps someone you know) know the answer to my other question on Cliffords?: quantumcomputing.stackexchange.com/questions/21993/… Nov 19 at 22:15