# Writing the notation when gates act on non successive registers

Suppose I have registers $$|a\rangle^{l}|b\rangle^{l} |c\rangle^{l}$$ and want an adder mod $$l$$ gate between the $$a$$ and $$c$$ registers. Let $$R$$ be the adder mod $$l$$ gate. So is this the correct notation for an operator $$U$$ that implements this $$U=R\otimes I_b^{\otimes l}.$$ But how do I convey that $$R$$ is between $$a$$ and $$c$$ and $$I$$ is for the register $$b$$?

Personally, I would just define $$R_{ac}$$ to be the unitary that acts $$R$$ between registers $$a$$ and $$c$$, and acts as identity everywhere else.

As always with notation there is not a "correct" way of doing things: it's just arbitrary conventions.

The most readable notation I see for your example involves separating the unitary $$R$$ into 2 virtual unitary matrices:

• $$R_a$$ the portion that acts on $$\vert a \rangle^l$$.
• $$R_c$$ the portion that acts on $$\vert c \rangle^l$$.

and "defining" $$R$$ as $$R = R_a \otimes R_c.$$

I called the matrices $$R_a$$ and $$R_c$$ "virtual unitaries because it is likely that they do not exist: the decomposition $$R = R_a \otimes R_c$$ will probably be impossible to compute because the matrix $$R$$ cannot be split as 2 separate transformations on $$\vert a \rangle^l$$ and $$\vert c \rangle^l$$.

Warning with this kind of non-standard notation: as the matrices involved are not really matrices (they are introduced just for the notation and might not exist), it may add more complexity/confusion than it helps.

In the end, your operation on $$\vert a \rangle^l\vert b \rangle^l\vert c \rangle^l$$ may be written as $$U = R_a \otimes I \otimes R_c.$$