# Kronecker notation of an operator

Suppose I have the state $$|A\rangle=|x\rangle^l\otimes |y\rangle^l \otimes |z\rangle^l \otimes |0\rangle_x^l\otimes |0\rangle_y^l\otimes |0\rangle_z^l$$. I perform the transformation between the $$|x\rangle$$ register and first $$0_x$$ register that is $$U|x,0\rangle\to |x,(2x)~\text{mod}~l\rangle$$ Suppose if $$|x\rangle=|x_0x_1x_2...x_{l-1}\rangle$$ and $$|0\rangle=|0_00_1000..0_{l-1}\rangle$$ then the result is $$|x\rangle|x_1x_2x_3....x_{l-1}0\rangle$$. So the tensorial notation for this operator on state $$A$$ is \begin{align} U=\prod_{i=0}^{l-2} \left( I^{\otimes 1+i}\otimes P_0\otimes I^{\otimes 3l-2}\otimes I\otimes I^{l-1-i}\otimes I^{\otimes 2l} \right. \\\left.+I^{\otimes 1+i}\otimes P_1\otimes I^{\otimes 3l-2}\otimes X\otimes I^{l-1-i}\otimes I^{\otimes 2l}\right) \end{align} Is this operator correct for the operation that I want to perform?

• Your function is not unitary, it wasn't supposed to be? – Mahathi Vempati Jun 2 '19 at 9:49