Solution
= Solution
Use $n$ <controlled-NOT gates> in parallel. For each $j=1,\ldots,n$, the $j$th input qubit is the control and the $j$th output-register qubit is the target. Their combined action is
$$
|x_1\cdots x_n\rangle|y_1\cdots y_n\rangle
\longmapsto
|x_1\cdots x_n\rangle
|y_1\mathbin\oplus x_1,\ldots,y_n\mathbin\oplus x_n\rangle,
$$
which is exactly
$$
\boxed{|x\rangle|y\rangle\mapsto|x\rangle|y\mathbin\oplus x\rangle=U_{\mathcal I}|x\rangle|y\rangle}.
$$