# What is the name for quantum gates that can be reversed?

Sorry if this question sounds too basic. But I can't find a word for those quantum gates than can be reversed (exchanging the target qubit with the control qubit) and still performs the same operation. Such as the CPhase or the SQSWAP.

I don't think they can be called "reversibles", because that means another thing.

Any ideas?

• "Swap-Covariant"? (I.e., they commute with the swap operation.) – Norbert Schuch Mar 10 at 19:00

symmetric?

Again, can be used in other contexts, but it is the right word for this context because what you're talking about is something that's invariant under swap. Perhaps the first time you use it, one should clarify with the phrase "symmetric under exchange of the inputs".

All quantum gates are reversible, in the sense that they always admit an inverse which is a quantum gate as well. This comes from the fact that quantum gates are (usually) required to be unitary.

What you probably meant to ask is how you call gates $$U$$ such that $$U^2=I$$. If you assume $$U$$ is unitary, that is $$U^\dagger U=UU^\dagger=I$$, then this is equivalent to the condition $$U^\dagger =U$$. Such gates are called Hermitian (or symmetric when they are real).

If, instead, you are referring to gates whose action is invariant under exchange of the inputs, then that can mean gates $$U$$ such that $$U|a,b\rangle=U|b,a\rangle$$ for all one-qubit states $$|a\rangle$$ and $$|b\rangle$$. This essentially means that the two middle columns of the matrix (in the standard choice of canonical basis) are the same. Yet another possible interpretation is that the matrix is required to have the same description when exchanging both the two input qubits and the two output qubits. This amounts to asking for $$U$$ to commute with the swap operation: $$[U,\mathrm{SWAP}]=0$$. You can compute explicitly the form of these matrices, but I don't think there is any universally accepted special name for these sorts of operations.

• That's what I thought was being asked when I first read the question. I don't think it is though. – DaftWullie Sep 5 '19 at 15:59
• @DaftWullie why not? "can be reversed and still perform the same operation" reads to me as asking for gates such that $U^{-1}=U^\dagger=U$ – glS Sep 5 '19 at 16:10
• but it is then clarified that the reversal means "exchanging the target qubit with the control qubit" – DaftWullie Sep 5 '19 at 16:11
• @MarkS you mean because it satisfies every single one of the conditions, or because I missed that it doesn't satisfy one of them? – glS Sep 5 '19 at 18:10
• @Alex personally I would find that a very confusing notation. "Symmetric" is used for matrices such that $A^T=A$. Something like symmetric under exchange of inputs might be better. – glS Sep 5 '19 at 20:59

One might consider this to be equivalent to the definition of a unitary transformation. All quantum gates are reversible, thus all quantum gates are unitary.

All quantum gates are unitary except measurement. quantum measurement isn't reversible.