While reading "Quantum Computation and Quantum Information" (by Nielsen and Chuang) I came across this line

A little thought shows that if we apply $U_f$ to the state $\vert x \rangle (\dfrac{\vert 0 \rangle - \vert 1 \rangle}{\sqrt{2}})$ then we obtain the state $(-1)^{f(x)} \vert x \rangle (\dfrac{\vert 0 \rangle - \vert 1 \rangle}{\sqrt{2}})$

Which was mentioned when explaining Deutsch's problem, where $\vert x, y \rangle$ is mapped to $\vert x, y \oplus f(x) \rangle$.

I was wondering, is this statement true for all unitary operations? Or is it limited to Deutsch's problem?


It is limited to matrix $U_f$ which maps $|x,y\rangle$ to $|x,y\oplus f(x)\rangle$, and the little thought is

$$U_f|x,-\rangle=\frac{1}{\sqrt{2}}(|x,0\oplus f(x)\rangle-|x,1\oplus f(x)\rangle)=$$ $$=\begin{cases} |x,-\rangle & \text{if }f(x)=0\\ -|x,-\rangle & \text{if }f(x)=1 \end{cases}=(-1)^{f(x)}|x,-\rangle$$ where $x\in\{0,1\}$ or generally $x\in\{0,1\}^n$


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