feat: ✨ add detailed explanation of the Deutsch Problem and Quantum Oracle in IQC 006

Author: gracefullight

blog/2026/03/30/iqc-006.md

@@ -37,9 +37,38 @@ x  & {\color{orange}{f_0}} & f_1 & f_2 & {\color{blue}{f_3}} & f_4 & {\color{blu
 \end{array}
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+## The Deutsch Problem
+
+- Given a boolean function $f:\{0,1\}^{n} \to \{0,1\}$, determine if $f$ is constant or balanced.
+- **Constant**: same output for all inputs.
+- **Balanced**: outpus 0 for half the inputs and 1 for the other half.
+
 ## The Quantum Oracle
 
 - Traditional **Oracle**: black box that computes $f$.
 - Query complexity: number of queries to the oracle needed to solve a problem.
 
 $$ x \xrightarrow{O_f} f(x) $$
+
+- **Quantum Oracle**: unitary operation that encodes $f$.
+
+$$ U_f \ket{x}\ket{y} = \ket{x}\ket{y \oplus f(x)} $$
+
+- First register: input $x$.
+- Second register: auxiliary qubit initialized to $\ket{0}$ or $\ket{1}$.
+- Oracle don't change $x$, but flips $y$ if $f(x)=1$.
+  - $0 \oplus 0 = 0$
+  - $0 \oplus 1 = 1$
+  - $1 \oplus 0 = 1$
+  - $1 \oplus 1 = 0$
+- $f(x) = 0$: $y$ unchanged.
+- $f(x) = 1$: $y$ flipped.
+- Example:
+  - $\ket{x}\ket{0}$
+  - $U_f \ket{x}\ket{0} = \ket{x}\ket{0} \oplus \ket{f(x)} = \ket{x}\ket{f(x)}$.
+  - if $f(x) = 0$: $\ket{x}\ket{0}$.
+  - if $f(x) = 1$: $\ket{x}\ket{1}$
+- so that it ignores the input $x$ and only flips the second qubit if $f(x)=1$.
+- it can be reversed: $(y \oplus f(x)) \oplus f(x) = y$.
+- if we put the second register to $\ket{-}$, $f(x)$ will be encoded **in the phase** of the first register:
+  - $\ket{x}\ket{-}\mapsto (-1)^{f(x)} \ket{x} \ket{-}$.