How to switch bit in the quantum state?

Question

I want to transform binary representation from one to other. Here, I have 2 registers one act as control register and another is target register with the same number of qubits $n$. Consider $n = 3$ qubits state and $N = 2^{n}$,

\begin{eqnarray} |\psi_{0}\rangle &=& |i\rangle \otimes \sum_{i = 0}^{N} a_{i}|i\rangle \\ |\psi_{0}\rangle &=& |i\rangle \otimes (a_{0}|000\rangle + a_{1}|001\rangle + a_{2}|010\rangle + a_{3}|011\rangle + a_{4}|100\rangle + a_{5}|101\rangle + a_{6}|110\rangle + a_{7}|111\rangle) \end{eqnarray}

I want a quantum circuit or algorithm ($T_{i}|\psi_{0}\rangle = |\psi_{1}\rangle$) to transform, for example, the target state, \begin{equation} |\psi_{1}\rangle = |101\rangle \otimes (a_{0}|000\rangle + a_{3}|001\rangle + a_{2}|010\rangle + a_{1}|011\rangle + a_{5}|100\rangle + a_{4}|101\rangle + a_{7}|110\rangle + a_{}|111\rangle). \end{equation} Please note that the amplitude $a_{i}$ still the same as before the operation. How to realize a quantum circuit like that, I know some special case where the operator will shift a bit to the right or left. However, I want a general form of operator $T$.

Answer 1

What you seem to be asking for is the general construction of an arbitrary permutation. There’s nothing quantum about this; the equivalent question in classical is how to construct an arbitrary reversible circuit of n bits. There are $2^n!$ such circuits, from which it is hopefully obvious that there is not a compact description for the vast majority.

I should also point out that you have a slight notational issue. For each of the states that you write, you’ve put a projector on the first term in the tensor product. This does not correspond to a state. You probably just meant the ket.

Answer 2

You can use a method described in Transformation of quantum states using uniformly controlled rotations.

Authors of the article introduced method how to change an arbitrary state $|a\rangle$ to state $|b\rangle$ with so-called uniformly controlled rotation, i.e. a rotation which rotational angle depends on combination of zeros and ones in controlling register.

The method uses $Ry$ and $Rz$ gates to achive this. $Ry$ gates are used for changing amplitudes and $Rz$ for changing phase. If you want to change only amplitudes, you can neglected part with $Rz$ gates.