Migrate task 1.3 from Measurements to Distinguishing States kata (#1489)

Migrating task 1.3 from the Measurements workbook to Distinguishing
States Kata.
This resolves a part of the [Issue
1185](#1185).

- Added folder plus_minus under distinguishing_states
- Added index, placeholder, solution, and verification under plus_minus
- Updated index file under distinguishing_states to contain the exercise
plus_minus

---------

Co-authored-by: Manvi-Agrawal <[email protected]>
Co-authored-by: Mariia Mykhailova <[email protected]>

katas/content/distinguishing_states/index.md

@@ -32,6 +32,14 @@ This kata is designed to get you familiar with the concept of measurements and u
     ]
 })
 
+@[exercise]({
+    "id": "distinguishing_states__plus_minus",
+    "title": "|+〉 or |-〉?",
+    "path": "./plus_minus/",
+    "qsDependencies": [
+        "../KatasLibrary.qs"
+    ]
+})
 
 @[section]({
     "id": "distinguishing_states__nonorthogonal_states",

katas/content/distinguishing_states/plus_minus/Placeholder.qs

@@ -0,0 +1,7 @@
+namespace Kata {
+    operation IsQubitPlus(q : Qubit) : Bool {
+        // Implement your solution here...
+
+        return false;
+    }
+}

katas/content/distinguishing_states/plus_minus/SolutionA.qs

@@ -0,0 +1,6 @@
+namespace Kata {
+    operation IsQubitPlus(q : Qubit) : Bool {
+        H(q);
+        return M(q) == Zero;
+    }
+}

katas/content/distinguishing_states/plus_minus/SolutionB.qs

@@ -0,0 +1,5 @@
+namespace Kata {
+    operation IsQubitPlus(q : Qubit) : Bool {
+        return Measure([PauliX], [q]) == Zero;
+    }
+}

katas/content/distinguishing_states/plus_minus/Verification.qs

@@ -0,0 +1,30 @@
+namespace Kata.Verification {
+    open Microsoft.Quantum.Katas;
+
+    operation StatePrep_IsQubitPlus(q : Qubit, state : Int) : Unit is Adj {
+        if state == 1 {
+            // convert |0⟩ to |+⟩
+            H(q);
+        } else {
+            // convert |0⟩ to |-⟩
+            X(q);
+            H(q);
+        }
+    }
+
+    @EntryPoint()
+    operation CheckSolution() : Bool {
+        let isCorrect = DistinguishTwoStates_SingleQubit(
+            StatePrep_IsQubitPlus,
+            Kata.IsQubitPlus,
+            ["|-⟩", "|+⟩"],
+            false
+        );
+        if isCorrect {
+            Message("Correct!");
+        } else {
+            Message("Incorrect.");
+        }
+        isCorrect
+    }
+}

katas/content/distinguishing_states/plus_minus/index.md

@@ -0,0 +1,3 @@
+**Input:** A qubit which is guaranteed to be in either the $\ket{+}$ or the $\ket{-}$ state.
+
+**Output:** `true` if the qubit was in the $\ket{+}$ state, or `false` if it was in the $\ket{-}$ state. The state of the qubit at the end of the operation does not matter.

katas/content/distinguishing_states/plus_minus/solution.md

@@ -0,0 +1,23 @@
+Both input states are superposition states, with equal absolute values of amplitudes of both basis states. This means if the sate is measured in the Pauli $Z$ basis, there is a 50-50 chance of measuring `One` or `Zero`, which won't give us the necessary information.
+
+To determine in which state the input qubit is with certainty, we want to transform the qubit into a state where there is no superposition with respect to the basis in which we perform the measurement.
+
+Consider how we can prepare the input states, starting with basis states: $H\ket{0} = \ket{+}$ and $H\ket{1} = \ket{-}$. This transformation can also be undone by applying the $H$ gate again (remember that the $H$ gate is self-adjoint, i.e., it equals its own inverse): $H\ket{+} = \ket{0}$ and $H\ket{-} = \ket{1}$.
+
+Once we have the $\ket{0}$ or $\ket{1}$ state, we can use the same principle as in previous task $\ket{0}$ or $\ket{1}$ to measure the state and report the outcome. Note that in this task return value `true` corresponds to input state $\ket{+}$, so we compare the measurement result with `Zero`.
+
+@[solution]({
+    "id": "distinguishing_states__plus_minus_solution_a",
+    "codePath": "SolutionA.qs"
+})
+
+#### Alternative solution
+
+Another possible solution could be to measure in the Pauli $X$ basis ($\ket{+}, \ket{-}$ basis), this means a transformation with the $H$ gate before measurement is not needed. Again, measurement result `Zero` would correspond to state $\ket{+}$.
+
+In Q#, measuring in another Pauli basis can be done with the `Measure()` operation.
+
+@[solution]({
+    "id": "distinguishing_states__plus_minus_solution_b",
+    "codePath": "SolutionB.qs"
+})