> For the complete documentation index, see [llms.txt](https://chiwei955201314.gitbook.io/quantum/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://chiwei955201314.gitbook.io/quantum/deutsch-jozsa/oracle-and-query.md).

# oracle and query

#### Oracle 

我們視oracle為一個黑盒，並不在意裡面如何運作，通常oracle會有一個輸入函式，而我們能做的就是詢問這個oracle，而oracle會告訴我們答案。

#### Query Model

我們在說的時間複雜度，基本上都是基於query的次數，oracle內的複雜度我們並不需要去討論，因為他是個黑盒。

給定一個輸入函數 $$f(x)$$ ，而根據query後的結果，oracle一般來說又可分為兩種：

$$
\begin{aligned}
&|x,w\rangle\rightarrow|x,w\oplus f(x)\rangle\\
&|x\rangle\rightarrow(-1)^{f(x)}|x\rangle
\end{aligned}
$$

我們常稱第二種為phase oracle，因為他改變了量子態的機率幅的正負號，而事實上，兩者是等價的，以下給個小小的證明：

給定一個 $$f(x)=\left{0,1\right}^n\rightarrow\left{0,1\right}$$ 將第二個register設定為 $$|-\rangle$$，並query

$$
\begin{aligned}
|x,-\rangle\rightarrow|x,-\oplus f(x)\rangle
\end{aligned}
$$

開始展開化簡，因 $$|-\rangle = \frac{|0\rangle-|1\rangle}{\sqrt{2}}$$ ，得到以下結果

$$
|x,-\oplus f(x)\rangle = \frac{|x\rangle|0\oplus f(x)\rangle-|x\rangle|1\oplus f(x)\rangle}{\sqrt{2}}
$$

仔細觀察可以發現，當 $$f(x)=0$$ ，化簡後與原來並無區別

$$
\begin{aligned}
|x,-\oplus 0\rangle
&= \frac{|x\rangle|0\oplus 0\rangle-|x\rangle|1\oplus 0\rangle}{\sqrt{2}}\\
&=\frac{|x\rangle|0\rangle-|x\rangle|1\rangle}{\sqrt{2}}\\
&=|x,-\rangle
\end{aligned}
$$

但是當 $$f(x)=1$$ ，卻會相差一個負號

$$
\begin{aligned}
|x,-\oplus 1\rangle
&= \frac{|x\rangle|0\oplus 1\rangle-|x\rangle|1\oplus 1\rangle}{\sqrt{2}}\\
&=\frac{|x\rangle|1\rangle-|x\rangle|0\rangle}{\sqrt{2}}\\
&=-|x,-\rangle
\end{aligned}
$$

透過這個證明，我們可以了解到兩種oracle的等價，也就是：

$$
|x,-\oplus f(x)\rangle = (-1)^{f(x)}|x\rangle|-\rangle
$$


---

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