Add task 2.1 to Superposition kata (#1395)

Co-authored-by: César Zaragoza Cortés <[email protected]>

katas/content/superposition/index.md

@@ -144,10 +144,21 @@ This kata is designed to get you familiar with the concept of superposition and
 })
 
 @[section]({
-    "id": "superposition__uneven_superpositions",
-    "title": "Uneven superpositions"
+    "id": "superposition__arbitrary_rotations",
+    "title": "Arbitrary Rotations"
 })
 
+@[exercise]({
+    "id": "superposition__unequal_superposition",
+    "title": "Unequal Superposition",
+    "path": "./unequal_superposition/",
+    "qsDependencies": [
+        "../KatasLibrary.qs",
+        "./Common.qs"
+    ]
+})
+
+
 @[section]({
     "id": "superposition__conclusion",
     "title": "Conclusion"

katas/content/superposition/unequal_superposition/Placeholder.qs

@@ -0,0 +1,6 @@
+namespace Kata {
+    operation UnequalSuperposition(q : Qubit, alpha : Double) : Unit {
+        // Implement your solution here...
+
+    }
+}

katas/content/superposition/unequal_superposition/Solution.qs

@@ -0,0 +1,5 @@
+namespace Kata {
+    operation UnequalSuperposition(q : Qubit, alpha : Double) : Unit {
+        Ry(2.0 * alpha, q);
+    }
+}

katas/content/superposition/unequal_superposition/Verification.qs

@@ -0,0 +1,28 @@
+namespace Kata.Verification {
+    open Microsoft.Quantum.Convert;
+    open Microsoft.Quantum.Katas;
+    open Microsoft.Quantum.Math;
+
+    operation UnequalSuperposition_Reference(q : Qubit, alpha : Double) : Unit is Adj + Ctl {
+        Ry(2.0 * alpha, q);
+    }
+
+    @EntryPoint()
+    operation CheckSolution() : Bool {
+        let limit = 36;
+        for i in 0 .. limit {
+            let alpha = 2.0 * PI() * IntAsDouble(i) / IntAsDouble(limit);
+            let solution = Kata.UnequalSuperposition(_, alpha);
+            let reference = UnequalSuperposition_Reference(_, alpha);
+            Message($"Testing for alpha = {alpha}...");
+            if not CheckOperationsEquivalenceOnZeroStateWithFeedback(
+                qs => solution(qs[0]), 
+                qs => reference(qs[0]),
+                1) {
+                return false;
+            }
+        }
+
+        true
+    }
+}

katas/content/superposition/unequal_superposition/index.md

@@ -0,0 +1,12 @@
+**Inputs:**
+
+1. A qubit in the $|0\rangle$ state.
+2. Angle $\alpha$, in radians, represented as `Double`.
+
+**Goal**: Change the state of the qubit to $\cos{α} |0\rangle + \sin{α} |1\rangle$.
+
+<details>
+  <summary><b>Need a hint?</b></summary>
+  Experiment with rotation gates you learned about in the Single-Qubit Gates kata.
+  Note that all rotation operators rotate the state by <i>half</i> of its angle argument.
+</details>

katas/content/superposition/unequal_superposition/solution.md

@@ -0,0 +1,20 @@
+We want to convert the $\ket{0}$ state to a parameterized superposition of $\ket{0}$ and $\ket{1}$, which suggests 
+that we are looking for some kind of a rotation operation. There are three main gates that implement rotations around various axes of the Bloch Sphere: 
+
+- $R_x(\theta) = \begin{bmatrix} \cos\frac{\theta}{2} & -i\sin\frac{\theta}{2} \\\ -i\sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{bmatrix}$
+- $R_y(\theta) = \begin{bmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{bmatrix}$
+- $R_z(\theta) = \begin{bmatrix} e^{-i\theta/2} & 0 \\\ 0 & e^{i\theta/2} \end{bmatrix}$
+
+If we were to apply the $R_x$ gate to a qubit in the $|0\rangle$ state, we would introduce complex coefficients to the amplitudes, which is clearly not what we're looking for. Similarly, the $R_z$ gate introduces only a global phase when applied to $|0\rangle$ state, so we can rule it out as well. This leaves only the $R_y$ as a starting point for the solution.
+
+Applying the $R_y$ gate to the $|0\rangle$ state, we get:
+$$R_y(\theta) |0\rangle = 
+\begin{bmatrix} \cos\frac{\theta}{2} & -\sin\frac{\theta}{2} \\\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2} \end{bmatrix} \begin{bmatrix} 1 \\\ 0 \end{bmatrix} = 
+\begin{bmatrix} \cos\frac{\theta}{2} \\\ \sin\frac{\theta}{2} \end{bmatrix} = \cos\frac{\theta}{2}|0\rangle + \sin\frac{\theta}{2}|1\rangle$$
+
+Therefore, applying the $R_y(2\alpha)$ gate to $|0\rangle$ is the solution to our problem. 
+
+@[solution]({
+    "id": "superposition__unequal_superposition_solution",
+    "codePath": "./Solution.qs"
+})