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Suppose I have the state $\frac{1}{2^l/2}\sum_{i=0}^{2^l-1}|0\rangle^{\otimes q}\otimes |i\rangle^{\otimes}|0\rangle_i^{\otimes l}$.

I perform some unitary transformation between the registers $|i\rangle^{\otimes}$ and $|0\rangle_i^{\otimes l} $.

That unitary operation is just the permutation, i.e. for each $|i\rangle^{\otimes l}|$ $0\rangle_i^{\otimes l}$ stores the permutation of $ |i\rangle^{\otimes l} $. Originally each $|0\rangle^{\otimes q}$ corresponded to each $ |i\rangle^{\otimes}$.

But now I want that $|0\rangle^{\otimes q}$ be associated to the permuted value stored in $ |0\rangle_i^{\otimes l}$ correspinding to $|i\rangle^{\otimes l}$.

For example initially I have the pairs $(5,0), (10,1), (45,2), (59,3)$ Now I perform the permutation on the second coordinate to obtain $0\to 1$, $1\to 3$, $2\to 0$, $ 3 \to 2$. So now I want pairs to be $ (5,1), (10,3), (45,0), (59,2)$.

How can this be obtained? Can someone help me out?

Edit: I give what i think of the solution. Consider an example of $4$ states as ignore the normalizing factor. $$|A\rangle= |5\rangle |0\rangle|0\rangle+|10\rangle |1\rangle|0\rangle+|45\rangle |2\rangle|0\rangle+|59\rangle |3\rangle|0\rangle$$ Now i apply the permutation on the second register to store the result in the third register to get $$|B\rangle= |5\rangle |0\rangle|1\rangle+|10\rangle |1\rangle|3\rangle+|45\rangle |2\rangle|0\rangle+|59\rangle |3\rangle|2\rangle$$ Now i swap the second and third qubits to get $$|C\rangle= |5\rangle |1\rangle|0\rangle+|10\rangle |3\rangle|1\rangle+|45\rangle |0\rangle|2\rangle+|59\rangle |2\rangle|3\rangle$$ Now since the third register is just the Hadamard on $2$ qubits, so i apply Hadamard on the third register of state $|C\rangle$ to get $$|D\rangle= |5\rangle |1\rangle|0\rangle+|10\rangle |3\rangle|0\rangle+|45\rangle |0\rangle|0\rangle+|59\rangle |2\rangle|0\rangle$$ Is this reasoning correct?

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  • $\begingroup$ Won't applying the permutation on the second register alone suffice? $\endgroup$ – Mahathi Vempati Jun 3 at 4:14

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