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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?

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

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