Consider some black-box that encodes some function \(f\) that is implemented as a controlled\(-U_f\) gate acting on the state space \(\mathcal{H}_A\otimes\mathcal{H}_B\) consisting of two registers \(A\) and \(B\) of qubits. Generally, when a controlled\(-U_f\) gate is applied it is thought of as only affecting the target register \(B\) leaving the control register \(A\) unchanged. Therefore, the action of the \(c-U_f\) gate is of the form

\[c-U_f: \left|x\right>\left|\phi\right>\mapsto \left|x\right>\left|\phi'\right>,\]

For some state \(\left|\phi'\right>\in\mathcal{H}_B\). Suppose some state \(\left|\psi\right>\in\mathcal{H}_B\) is chosen to satisfy

\[c-U_f(\left|x\right>\left|\psi\right>)= v_x\left|x\right>\left|\psi\right>,\]

where \(v_x\) is some complex number that could depends on \(x\). In this case, the state \(\left|x\right>\left|\psi\right>\) is called an

\[c-U_f(\left|y\right>\left|\psi\right>)= v_y\left|y\right>\left|\psi\right>,\]

for some other state \(\left|y\right>\in\mathcal{H}_B\) and eigenvalue \(v_y\).

This situation is more interesting when the control register is in some superposition of states such as \(\alpha\left|x\right>+\beta\left|y\right>\in\mathcal{H}_A\). Then

\[\begin{array}{r l}

c-U_f\big((\alpha\left|x\right>+\beta\left|y\right>)\left|\psi\right>\big) & =c-U_f(\alpha\left|x\right>\left|\psi\right>)+c-U_f(\beta\left|y\right>\left|\psi\right>) \\

&= v_x\alpha\left|x\right>\left|\psi\right>+v_y\beta\left|y\right>\left|\psi\right> \\

&=\big(v_x\alpha\left|x\right>+v_y\beta\left|y\right>\big)\left|\psi\right>,

\end{array}\]

and depending on the eigenvalues the state in the control register may contain a relative phase factor between the states \(\left|x\right>\) and \(\left|y\right>\). In this context, by associating the eigenvalue to the state in the control register, the action of the controlled\(-U_f\) gate can effectively be thought of as changing the state in the control register instead of the target register. Then by performing a measurement on the states, the existence of this phase factor may contain relevant information about the function \(f\) encoded in the black box.

This technique of preparing the input in an eigenstate of a controlled\(-U\) gate, and then associating the eigenvalue to the control register is referred to as

\[c-U_f: \left|x\right>\left|\phi\right>\mapsto \left|x\right>\left|\phi'\right>,\]

For some state \(\left|\phi'\right>\in\mathcal{H}_B\). Suppose some state \(\left|\psi\right>\in\mathcal{H}_B\) is chosen to satisfy

\[c-U_f(\left|x\right>\left|\psi\right>)= v_x\left|x\right>\left|\psi\right>,\]

where \(v_x\) is some complex number that could depends on \(x\). In this case, the state \(\left|x\right>\left|\psi\right>\) is called an

**eigenstate**of the operator \(c-U_f\), and \(v_x\) is called the**eigenvalue**of the**eigenstate**. Moreover, the state \(\left|\psi\right>\) may also satisfy\[c-U_f(\left|y\right>\left|\psi\right>)= v_y\left|y\right>\left|\psi\right>,\]

for some other state \(\left|y\right>\in\mathcal{H}_B\) and eigenvalue \(v_y\).

This situation is more interesting when the control register is in some superposition of states such as \(\alpha\left|x\right>+\beta\left|y\right>\in\mathcal{H}_A\). Then

\[\begin{array}{r l}

c-U_f\big((\alpha\left|x\right>+\beta\left|y\right>)\left|\psi\right>\big) & =c-U_f(\alpha\left|x\right>\left|\psi\right>)+c-U_f(\beta\left|y\right>\left|\psi\right>) \\

&= v_x\alpha\left|x\right>\left|\psi\right>+v_y\beta\left|y\right>\left|\psi\right> \\

&=\big(v_x\alpha\left|x\right>+v_y\beta\left|y\right>\big)\left|\psi\right>,

\end{array}\]

and depending on the eigenvalues the state in the control register may contain a relative phase factor between the states \(\left|x\right>\) and \(\left|y\right>\). In this context, by associating the eigenvalue to the state in the control register, the action of the controlled\(-U_f\) gate can effectively be thought of as changing the state in the control register instead of the target register. Then by performing a measurement on the states, the existence of this phase factor may contain relevant information about the function \(f\) encoded in the black box.

This technique of preparing the input in an eigenstate of a controlled\(-U\) gate, and then associating the eigenvalue to the control register is referred to as

**phase kick-back**. A whole class of quantum algorithms exploits this technique in order to learn some property of a function \(f\) encoded in a black-box with fewer queries than would be needed in the classical case.
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