small tweaks to lecture 9 material for clarity and correctness

lectures/phase-ii/lecture9.tex

@@ -24,7 +24,7 @@ \subsection*{Problem Statement}
 and only one item is “special.” Classically, you must check each item one by
 one (on average, \( O(2^n) \) trials) to find the special item. Grover's
 algorithm, however, uses quantum amplitude amplification to solve this
-problem in only \( O(\sqrt{2^n}) \) iterations- a quadratic speedup over
+problem in only \( O(\sqrt{2^n}) \) iterations—a quadratic speedup over
 classical search.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -57,8 +57,8 @@ \subsection*{Grover's Algorithm Circuit}
           O \ket{x} = (-1)^{f(x)} \ket{x}.
         \]
 
-        In our example, if the winning state is \(\ket{w}\), then \( O\ket{w}
-        = -\ket{w} \).
+        For example, if the winning state is \(\ket{w}\), then
+        \( O\ket{w} = -\ket{w} \).
 
       \item \textbf{Diffusion Operator \(D\):} Reflect all amplitudes about
         the average amplitude. This operator is given by
@@ -67,18 +67,20 @@ \subsection*{Grover's Algorithm Circuit}
           D = 2\ket{s}\bra{s} - I.
         \]
 
-        In practice, \( D \) is implemented by the sequence:
+        In practice, \( D \) is implemented as
 
         \[
-          H^{\otimes n} \; X^{\otimes n} \; (CZ) \; X^{\otimes n} \;
-          H^{\otimes n}.
+          D = H^{\otimes n} \; X^{\otimes n} \; (CZ) \; X^{\otimes n} \;
+          H^{\otimes n},
         \]
 
+        where the \(CZ\) gate here represents a controlled phase flip on
+        \(\ket{1}^{\otimes n}\) (for \(n>2\), this is a multi-controlled \(Z\)
+        gate).
     \end{itemize}
 \end{enumerate}
 
-The following diagram shows the high-level circuit for an \( n \)-qubit
-Grover algorithm:
+The high-level circuit for an \( n \)-qubit Grover algorithm is illustrated as:
 
 \[
 \begin{quantikz}
@@ -100,14 +102,13 @@ \subsection*{Grover's Algorithm Circuit}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsubsection*{2-Qubit Example}
 
-To build intuition, let us consider the case \( n=2 \) (i.e. \( N=4 \)) with
-the winning state chosen as \(\ket{11}\).
+To build intuition, consider the case \( n=2 \) (i.e., \( N=4 \)) with the winning
+state chosen as \(\ket{11}\).
 
 \vspace{0.3cm}
 
-\textbf{Oracle:} For a 2-qubit system, the Oracle \(O\) can be implemented as
-a controlled-Z (CZ) gate since
-
+\textbf{Oracle:} For a 2-qubit system, the Oracle \(O\) can be implemented as a
+controlled-\(Z\) (CZ) gate:
 \[
   CZ =
   \begin{pmatrix}
@@ -128,10 +129,9 @@ \subsubsection*{2-Qubit Example}
   D = H^{\otimes 2}\; X^{\otimes 2}\; CZ\; X^{\otimes 2}\; H^{\otimes 2}.
 \]
 
-\textbf{2-Qubit Grover Circuit:}
-
+\vspace{0.3cm}
 
-The following circuit implements Grover's algorithm for 2 qubits:
+\textbf{2-Qubit Grover Circuit:}
 
 \[
 \begin{quantikz}
@@ -146,13 +146,11 @@ \subsubsection*{2-Qubit Example}
 \textbf{Explanation:}
 
 \begin{itemize}
-  \item \textbf{Oracle:} The Oracle adds a phase of \(-1\) to the winner
+  \item \textbf{Oracle:} The Oracle adds a phase of \(-1\) to the winning
     state \(\ket{11}\), effectively marking it.
-
-  \item \textbf{Diffusion Operator:} The diffusion circuit reflects all
-    amplitudes about the average. This inversion about the mean amplifies the
-    amplitude of the marked state while reducing that of all others.
-
+  \item \textbf{Diffusion Operator:} The diffusion circuit (implemented as
+    \(H\;X\;CZ\;X\;H\)) reflects all amplitudes about the average. This
+    inversion about the mean amplifies the amplitude of the marked state.
 \end{itemize}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -188,10 +186,10 @@ \subsection*{Walkthrough of the Algorithm and Generalization to \(n\) Qubits}
     \[
       O\ket{x} = (-1)^{f(x)}\ket{x},
     \]
+    (i.e., flip the phase of the marked state).\;
 
-    \quad (i.e., flip the phase of the marked state).\;
-    Apply the Diffusion Operator \( D = 2\ket{s}\bra{s} - I \) (which
-    reflects amplitudes about the average).\;
+    Apply the Diffusion Operator \( D = 2\ket{s}\bra{s} - I \)
+    (which reflects amplitudes about the average).\;
   }
 
   Measure the state in the computational basis to obtain \( x_0 \).\;
@@ -211,15 +209,13 @@ \subsection*{Walkthrough of the Algorithm and Generalization to \(n\) Qubits}
     \(O\ket{w} = -\ket{w}\)).
 
   \item \textbf{Diffusion (Inversion about the Mean):} This operator reflects
-    the state vector about the average amplitude. Geometrically, if you
-    visualize the amplitudes as vectors in the complex plane, the Oracle
-    rotates the winning state's amplitude by 180°, and the diffusion operator
-    then “pushes” this amplitude further away from the average, thereby
-    amplifying it.
+    the state vector about the average amplitude. Geometrically, one can
+    view the process as a rotation in a two-dimensional subspace (see
+    below).
 
   \item \textbf{Iteration:} Repeating the Oracle and Diffusion steps
-    approximately \( \sqrt{2^n} \) times ensures that the probability
-    amplitude of the marked state is maximized.
+    approximately \( \sqrt{2^n} \) times amplifies the probability amplitude of
+    the marked state.
 
   \item \textbf{Measurement:} Finally, a measurement in the computational
     basis yields the marked element with high probability.