Migrate nonlocal games 1596 task1 - quantum (#1745)

The link to the issue: #1596

This pull request is covering
1. CHSH quantum tasks (2.2, 2.4; 2.5 is converted to a demo)
2. CHSH discussion
3. Few changes in CHSH classical

---------

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

katas/content/nonlocal_games/chsh_classical_strategy/Verification.qs

@@ -6,8 +6,8 @@ namespace Kata.Verification {
     operation CheckSolution() : Bool {
         mutable wins = 0;
         for i in 1..1000 {
-            let x = DrawRandomInt(0, 1) == 1 ? true | false;
-            let y = DrawRandomInt(0, 1) == 1 ? true | false;
+            let x = DrawRandomBool(0.5);
+            let y = DrawRandomBool(0.5);
             let (a, b) = (Kata.AliceClassical(x), Kata.BobClassical(y));
             if ((x and y) == (a != b)) {
                 set wins = wins + 1;

katas/content/nonlocal_games/chsh_classical_win_condition/index.md

@@ -1,6 +1,7 @@
 **Input:** 
-* Alice and Bob's starting bits (X and Y).
-* Alice and Bob's output bits (A and B).
+
+- Alice and Bob's starting bits (X and Y).
+- Alice and Bob's output bits (A and B).
 
 **Output:**
   True if Alice and Bob won the CHSH game, that is, if X ∧ Y = A ⊕ B, and false otherwise.

katas/content/nonlocal_games/chsh_quantum_alice_strategy/Placeholder.qs

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

katas/content/nonlocal_games/chsh_quantum_alice_strategy/Solution.qs

@@ -0,0 +1,12 @@
+namespace Kata {
+    operation AliceQuantum (bit : Bool, qubit : Qubit) : Bool {
+        if bit {
+            let q = MResetX(qubit);
+            return q == One;
+        }
+        else {
+            let q = MResetZ(qubit);
+            return q == One;
+        }
+    }
+}

katas/content/nonlocal_games/chsh_quantum_alice_strategy/Verification.qs

@@ -0,0 +1,49 @@
+namespace Kata.Verification {
+    open Microsoft.Quantum.Convert;
+    open Microsoft.Quantum.Random;
+
+    @EntryPoint()
+    operation CheckSolution() : Bool {
+        use q = Qubit();
+        for _ in 1 .. 4 {
+            // repeat 4 times since we are testing a measurement and wrong basis still might get
+            // the correct answer, reduces probability of false positives
+            let result = Kata.AliceQuantum(false, q);
+            Reset(q);
+            if (result != false) {
+                Message("Measuring |0⟩ in the Z basis returned incorrect value; expected false");
+                return false;
+            }
+
+            // apply the Pauli X gate
+            X(q);
+            let result = Kata.AliceQuantum(false, q);
+            Reset(q);
+            if (result != true) {
+                Message("Measuring |1⟩ in the Z basis returned incorrect value; expected true");
+                return false;
+            }
+
+            // apply the Hadamard gate
+            H(q);
+            let result = Kata.AliceQuantum(true, q);
+            Reset(q);
+            if (result != false) {
+                Message("Measuring |+⟩ in the X basis returned incorrect value; expected false");
+                return false;
+            }
+
+            // apply the Pauli X and then the Hadamard gate
+            X(q);
+            H(q);
+            let result = Kata.AliceQuantum(true, q);
+            Reset(q);
+            if (result != true) {
+                Message("Measuring |-⟩ in the X basis returned incorrect value; expected true");
+                return false;
+            }
+        }
+        Message("Correct!");
+        true
+    }
+}

katas/content/nonlocal_games/chsh_quantum_alice_strategy/index.md

@@ -0,0 +1,12 @@
+In the quantum version of the game, the players still can not communicate during the game,
+but they are allowed to share qubits from a Bell pair before the start of the game.
+
+**Inputs:**
+
+- Alice's starting bit (X).
+- Alice's half of Bell pair she shares with Bob.
+
+**Goal:**
+  Measure Alice's qubit in the Z basis if her bit is 0 (false), or the X basis if her bit is 1 (true) and
+  return the measurement result as Boolean value: map `Zero` to false and `One` to true.
+  The state of the qubit after the operation does not matter.

katas/content/nonlocal_games/chsh_quantum_alice_strategy/solution.md

@@ -0,0 +1,12 @@
+In Q#, you can perform measurements in a specific basis using either the
+[Measure operation](https://learn.microsoft.com/qsharp/api/qsharp-lang/microsoft.quantum.intrinsic/measure)
+or convenient shorthands for measure-and-reset-to-$\ket{0}$ sequence of operations
+[MResetZ](https://learn.microsoft.com/qsharp/api/qsharp-lang/microsoft.quantum.measurement/mresetz) and
+[MResetX](https://learn.microsoft.com/qsharp/api/qsharp-lang/microsoft.quantum.measurement/mresetx).
+
+(See the the lesson below for details on why Alice should follow this strategy.)
+
+@[solution]({
+    "id": "nonlocal_games__chsh_quantum_alice_strategy_solution",
+    "codePath": "Solution.qs"
+})

katas/content/nonlocal_games/chsh_quantum_bob_strategy/Placeholder.qs

@@ -0,0 +1,9 @@
+namespace Kata {
+    open Microsoft.Quantum.Math;
+
+    operation BobQuantum (bit : Bool, qubit : Qubit) : Bool {
+        // Implement your solution here...
+
+        return false;
+    }
+}

katas/content/nonlocal_games/chsh_quantum_bob_strategy/Solution.qs

@@ -0,0 +1,9 @@
+namespace Kata {
+    open Microsoft.Quantum.Math;
+
+    operation BobQuantum (bit : Bool, qubit : Qubit) : Bool {
+        let angle = 2.0 * PI() / 8.0;
+        Ry(not bit ? -angle | angle, qubit);
+        return M(qubit) == One;
+    }
+}

katas/content/nonlocal_games/chsh_quantum_bob_strategy/Verification.qs

@@ -0,0 +1,55 @@
+namespace Kata.Verification {
+    open Microsoft.Quantum.Math;
+
+    operation RotateBobQubit (clockwise : Bool, qubit : Qubit) : Unit {
+        if (clockwise) {
+            Ry(-PI()/4.0, qubit);
+        } else {
+            Ry(PI()/4.0, qubit);
+        }
+    }
+
+    @EntryPoint()
+    operation CheckSolution() : Bool {
+        for _ in 1 .. 4 {
+            // repeat 4 times since we are testing a measurement and wrong basis still might get
+            // the correct answer, reduces probability of false positives
+            use q = Qubit();
+            RotateBobQubit(false, q);
+            let result = Kata.BobQuantum(false, q);
+            Reset(q);
+            if (result != false) {
+                Message("π/8 from |0⟩ not measured as false");
+                return false;
+            }
+
+            X(q);
+            RotateBobQubit(false, q);
+            let result = Kata.BobQuantum(false, q);
+            Reset(q);
+            if (result != true) {
+                Message("π/8 from |1⟩ not measured as true");
+                return false;
+            }
+
+            RotateBobQubit(true, q);
+            let result = Kata.BobQuantum(true, q);
+            Reset(q);
+            if (result != false) {
+                Message("-π/8 from |0⟩ not measured as false");
+                return false;
+            }
+
+            X(q);
+            RotateBobQubit(true, q);
+            let result = Kata.BobQuantum(true, q);
+            Reset(q);
+            if (result != true) {
+                Message("-π/8 from |1⟩ not measured as true");
+                return false;
+            }
+        }
+        Message("Correct!");
+        true
+    }   
+}

katas/content/nonlocal_games/chsh_quantum_bob_strategy/index.md

@@ -0,0 +1,11 @@
+**Inputs:**
+
+- Bob's starting bit (Y).
+- Bob's half of Bell pair he shares with Alice.
+
+**Goal:**
+Measure Bob's qubit in the $\frac{\pi}{8}$ basis if his bit is 0 (false), or the $-\frac{\pi}{8}$ basis
+if his bit is 1 (true) and return the measurement result as a Boolean value: map `Zero` to false and `One` to true.
+The state of the qubit after the operation does not matter.
+
+Measuring a qubit in the $\theta$ basis is the same as rotating the qubit by $\theta$, clockwise, and then making a standard measurement in the Z basis.

katas/content/nonlocal_games/chsh_quantum_bob_strategy/solution.md

@@ -0,0 +1,11 @@
+A suitable $R_y$ rotation can be used to go from the computational basis $\{ \ket{0}, \ket{1} \}$ to the $\{ \ket{\psi_+}, \ket{\psi_-} \}$ basis and vice versa.
+
+To implement the described transformation in Q#, we need to rotate the qubit by $\frac{\pi}{8}$ clockwise if `bit == false` or counterclockwise if `bit == true` and then perform a measurement.
+We can do the rotation using the $R_y$ gate (note the negation of the Boolean parameter we need to do).
+
+(See the lesson below for details on why Bob should follow this strategy.)
+
+@[solution]({
+    "id": "nonlocal_games__chsh_quantum_bob_strategy_solution",
+    "codePath": "Solution.qs"
+})

katas/content/nonlocal_games/examples/CHSHGameDemo.qs

@@ -0,0 +1,52 @@
+namespace Quantum.Kata.CHSHGame {
+    open Microsoft.Quantum.Random;
+    open Microsoft.Quantum.Math;
+    open Microsoft.Quantum.Convert;
+
+    function WinCondition (x : Bool, y : Bool, a : Bool, b : Bool) : Bool {
+        return (x and y) == (a != b);
+    }
+
+    function AliceClassical (x : Bool) : Bool {
+        return false;
+    }
+
+    function BobClassical (y : Bool) : Bool {
+        return false;
+    }
+
+    operation AliceQuantum (bit : Bool, qubit : Qubit) : Bool {
+        if bit {
+            return MResetX(qubit) == One;
+        }
+        return MResetZ(qubit) == One;
+    }
+
+    operation BobQuantum (bit : Bool, qubit : Qubit) : Bool {
+        let angle = 2.0 * PI() / 8.0;
+        Ry(not bit ? -angle | angle, qubit);
+        return M(qubit) == One;
+    }
+
+    @EntryPoint()
+    operation CHSH_GameDemo() : Unit {
+        use (aliceQubit, bobQubit) = (Qubit(), Qubit());
+        mutable classicalWins = 0;
+        mutable quantumWins = 0;
+        let iterations = 1000;
+        for _ in 1 .. iterations {
+            H(aliceQubit);
+            CNOT(aliceQubit, bobQubit);
+            let (x, y) = (DrawRandomBool(0.5), DrawRandomBool(0.5));
+            if WinCondition(x, y, AliceClassical(x), BobClassical(y)) {
+                set classicalWins += 1;
+            }
+            if WinCondition(x, y, AliceQuantum(x, aliceQubit), BobQuantum(y, bobQubit)) {
+                set quantumWins += 1;
+            }
+            ResetAll([aliceQubit, bobQubit]);
+        }
+        Message($"Percentage of classical wins is {100.0*IntAsDouble(classicalWins)/IntAsDouble(iterations)}%");
+        Message($"Percentage of quantum wins is {100.0*IntAsDouble(quantumWins)/IntAsDouble(iterations)}%");
+    }
+}