Migrate nonlocal games: GHZ game classical (#1783)

The link to the issue: #1596

This pull request is covering

1. GHZ classic tasks 1.1
2. GHZ task 1.2 (random) is dropped. Task 1.3 is expanded into
implementation of 3 strategies (Alice, Bob and Charlie)
This approach allow to create more complicated output like [true, false,
false] or [r, DrawRandomBool(0.8), not t]
3. GHZ classic task 1.4

---------

Co-authored-by: Mariia Mykhailova <michaylova@gmail.com>

katas/content/nonlocal_games/chsh_classical_strategy/solution.md

@@ -1,5 +1,5 @@
-If Alice and Bob always return TRUE, they will have a 75% win rate, since TRUE ⊕ TRUE = FALSE, and the AND operation on their input bits will be FALSE with 75% probability.
-Alternatively, Alice and Bob could agree to always return FALSE to achieve the same 75% win probability. A classical strategy cannot achieve a higher success probability.
+If Alice and Bob always return TRUE, they will have a $75\%$ win rate, since TRUE $\oplus$ TRUE == FALSE, and the AND operation on their input bits will be FALSE with $75\%$ probability.
+Alternatively, Alice and Bob could agree to always return FALSE to achieve the same $75\%$ win probability. A classical strategy cannot achieve a higher success probability.
 
 @[solution]({
     "id": "nonlocal_games__chsh_classical_strategy_solution",

katas/content/nonlocal_games/chsh_classical_win_condition/index.md

@@ -1,8 +1,8 @@
-**Input:** 
+**Inputs:** 
 
 - 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.
+  True if Alice and Bob won the CHSH game, that is, if X $\land$ Y = A $\oplus$ B, and false otherwise.
 

katas/content/nonlocal_games/chsh_classical_win_condition/solution.md

@@ -1,7 +1,10 @@
-There are four input pairs (X, Y) possible, (0,0), (0,1), (1,0), and (1,1), each with 25% probability.
+There are four input pairs (X, Y) possible, (0,0), (0,1), (1,0), and (1,1), each with $25\%$ probability.
 In order to win, Alice and Bob have to output different bits if the input is (1,1), and same bits otherwise.
 
-To check whether the win condition holds, you need to compute $x ∧ y$ and $a ⊕ b$ and to compare these values: if they are equal, Alice and Bob won. You can compute these values using [built-in operators](https://learn.microsoft.com/azure/quantum/user-guide/language/expressions/logicalexpressions): $x ∧ y$ as `x and y` and $a ⊕ b$ as `a != b`.
+To check whether the win condition holds, you need to compute X $\land$ Y and A $\oplus$ B and to compare these values:
+if they are equal, Alice and Bob won. You can compute these values using
+[built-in operators](https://learn.microsoft.com/azure/quantum/user-guide/language/expressions/logicalexpressions):
+X $\land$ Y as `x and y` and A $\oplus$ B as `a != b`.
 
 
 @[solution]({

katas/content/nonlocal_games/ghz_classical_game/Placeholder.qs

@@ -0,0 +1,7 @@
+namespace Kata {
+    operation PlayClassicalGHZ (strategies : (Bool => Bool)[], inputs : Bool[]) : Bool[] {
+        // Implement your solution here...
+
+        return [];
+    }
+}

katas/content/nonlocal_games/ghz_classical_game/Solution.qs

@@ -0,0 +1,11 @@
+namespace Kata {
+    operation PlayClassicalGHZ (strategies : (Bool => Bool)[], inputs : Bool[]) : Bool[] {
+        let r = inputs[0];
+        let s = inputs[1];
+        let t = inputs[2];
+        let a = strategies[0](r);
+        let b = strategies[1](s);
+        let c = strategies[2](t);
+        return [a, b, c];
+    }
+}

katas/content/nonlocal_games/ghz_classical_game/Verification.qs

@@ -0,0 +1,50 @@
+namespace Kata.Verification {
+
+    // All possible starting bits (r, s and t) that the referee can give
+    // to Alice, Bob and Charlie.
+    function RefereeBits () : Bool[][] {
+        return [[false, false, false],
+                [true, true, false],
+                [false, true, true],
+                [true, false, true]];
+    }
+
+    operation TestStrategy (input : Bool, mode : Int) : Bool {
+        return mode == 0 ? false | mode == 1 ? true | mode == 2 ? input | not input;
+    }
+
+    operation PlayClassicalGHZ_Reference (strategies : (Bool => Bool)[], inputs : Bool[]) : Bool[] {
+        let r = inputs[0];
+        let s = inputs[1];
+        let t = inputs[2];
+        let a = strategies[0](r);
+        let b = strategies[1](s);
+        let c = strategies[2](t);
+        return [a, b, c];
+    }
+
+    @EntryPoint()
+    operation CheckSolution() : Bool {
+        let inputs = RefereeBits();
+        for rst in inputs {
+            // To test the interaction, run it on deterministic strategies (not necessarily good ones)
+            // This logic helps to detect errors in user PlayClassicalGHZ implementation like
+            // using the wrong sequence of output bits or not using the strategies at all. 
+            for mode_1 in 0 .. 3 {
+                for mode_2 in 0 .. 3 {
+                    for mode_3 in 0 .. 3 {
+                        let strategies = [TestStrategy(_, mode_1), TestStrategy(_, mode_2), TestStrategy(_, mode_3)];
+                        let actual = Kata.PlayClassicalGHZ(strategies, rst);
+                        let expected = PlayClassicalGHZ_Reference(strategies, rst);
+                        if actual != expected {
+                            Message($"Expected {expected}, got {actual} for {rst}");
+                            return false;
+                        }
+                    }
+                }
+            }
+        }
+        Message("Correct!");
+        true
+    }
+}

katas/content/nonlocal_games/ghz_classical_game/index.md

@@ -0,0 +1,8 @@
+**Inputs:**
+
+1. An array of three operations which implement the classical strategies of the players (that is, take an input bit and produce an output bit),
+2. An array of 3 input bits that should be passed to the players.
+
+**Output:**
+
+An array of three bits that will be produced if each player uses their given strategy.

katas/content/nonlocal_games/ghz_classical_game/solution.md

@@ -0,0 +1,6 @@
+You are given both the input bits and the strategy each of the players are using, so you have simply to convert them to the output bits and return those.
+
+@[solution]({
+    "id": "nonlocal_games__ghz_classical_game_solution",
+    "codePath": "Solution.qs"
+})

katas/content/nonlocal_games/ghz_classical_strategy/Placeholder.qs

@@ -0,0 +1,19 @@
+namespace Kata {
+    operation AliceClassical (r : Bool) : Bool {
+        // Implement your solution here...
+
+        return false;
+    }
+
+    operation BobClassical (s : Bool) : Bool {
+        // Implement your solution here...
+
+        return false;
+    }
+
+    operation CharlieClassical (t : Bool) : Bool {
+        // Implement your solution here...
+
+        return false;
+    }
+}

katas/content/nonlocal_games/ghz_classical_strategy/Solution.qs

@@ -0,0 +1,14 @@
+namespace Kata {
+    operation AliceClassical (r : Bool) : Bool {
+        return true;
+    }
+
+    operation BobClassical (s : Bool) : Bool {
+        return true;
+    }
+
+    operation CharlieClassical (t : Bool) : Bool {
+        return true;
+    }
+}
+

katas/content/nonlocal_games/ghz_classical_strategy/Verification.qs

@@ -0,0 +1,53 @@
+namespace Kata.Verification {
+    open Microsoft.Quantum.Arrays;
+    open Microsoft.Quantum.Convert;
+    open Microsoft.Quantum.Logical;    
+    open Microsoft.Quantum.Random;
+
+    function WinCondition_Reference (rst : Bool[], abc : Bool[]) : Bool {
+        return (rst[0] or rst[1] or rst[2]) == (abc[0] != abc[1] != abc[2]);
+    }
+
+    // All possible starting bits (r, s and t) that the referee can give
+    // to Alice, Bob and Charlie.
+    function RefereeBits () : Bool[][] {
+        return [[false, false, false],
+                [true, true, false],
+                [false, true, true],
+                [true, false, true]];
+    }
+
+    operation PlayClassicalGHZ_Reference (strategies : (Bool => Bool)[], inputs : Bool[]) : Bool[] {
+        let r = inputs[0];
+        let s = inputs[1];
+        let t = inputs[2];
+        let a = strategies[0](r);
+        let b = strategies[1](s);
+        let c = strategies[2](t);
+        return [a, b, c];
+    }
+
+    @EntryPoint()
+    operation CheckSolution() : Bool {
+        let inputs = RefereeBits();
+        let strategies = [Kata.AliceClassical, Kata.BobClassical, Kata.CharlieClassical];
+
+        let iterations = 1000;
+        mutable wins = 0;
+        for _ in 0 .. iterations - 1 {
+            for bits in inputs {
+                let abc = PlayClassicalGHZ_Reference(strategies, bits);
+                if WinCondition_Reference(bits, abc) {
+                    set wins = wins + 1;
+                }
+            }
+        }
+        // The solution is correct if the players win 75% (3/4) of the time.
+        if wins < iterations*Length(inputs)*3/4 {
+            Message($"Alice, Bob, and Charlie's classical strategy gets {wins} wins out of {iterations*Length(inputs)} possible inputs, which is not optimal");
+            return false;
+        }
+        Message("Correct!");
+        true
+    }
+}

katas/content/nonlocal_games/ghz_classical_strategy/index.md

@@ -0,0 +1,11 @@
+In this task you have to implement three functions, one for each player's classical strategy.
+Note that they are covered by one test, so you have to implement all of them to pass the test.
+In each function, the input is the starting bit of the corresponding player, and it should return the output bit chosen by that player.
+
+**Input:**
+
+Alice, Bob, and Charlie's starting bits (R, S, and T).
+
+**Output:**
+
+Alice, Bob, and Charlie's output bits (A, B, and C) to maximize their chance of winning.

katas/content/nonlocal_games/ghz_classical_strategy/solution.md

@@ -0,0 +1,11 @@
+If all three players return TRUE, then A $\oplus$ B $\oplus$ C == TRUE by necessity (since the sum of their bits is odd).
+This will win against inputs 011, 101, and 110 and lose against 000.
+Another solution is one player retuns TRUE, and two others return FALSE.
+
+Since the four above inputs have equal probability, and represent all possible inputs,
+either of these deterministic strategies wins with $75\%$ probability.
+
+@[solution]({
+    "id": "nonlocal_games__ghz_classical_strategy_solution",
+    "codePath": "Solution.qs"
+})

katas/content/nonlocal_games/ghz_win_condition/Placeholder.qs

@@ -0,0 +1,7 @@
+namespace Kata {
+    function WinCondition (rst : Bool[], abc : Bool[]) : Bool {
+        // Implement your solution here...
+
+        return false;
+    }
+}

katas/content/nonlocal_games/ghz_win_condition/Solution.qs

@@ -0,0 +1,5 @@
+namespace Kata {
+    function WinCondition (rst : Bool[], abc : Bool[]) : Bool {
+        return (rst[0] or rst[1] or rst[2]) == (abc[0] != abc[1] != abc[2]);
+    }
+}

katas/content/nonlocal_games/ghz_win_condition/Verification.qs

@@ -0,0 +1,34 @@
+namespace Kata.Verification {
+    open Microsoft.Quantum.Convert;
+
+    function WinCondition_Reference (rst : Bool[], abc : Bool[]) : Bool {
+        return (rst[0] or rst[1] or rst[2]) == (abc[0] != abc[1] != abc[2]);
+    }
+
+    // All possible starting bits (r, s and t) that the referee can give
+    // to Alice, Bob and Charlie.
+    function RefereeBits () : Bool[][] {
+        return [[false, false, false],
+                [true, true, false],
+                [false, true, true],
+                [true, false, true]];
+    }
+
+    @EntryPoint()
+    function CheckSolution() : Bool {
+        for rst in RefereeBits() {
+            for i in 0 .. 1 <<< 3 - 1 {
+                let abc = IntAsBoolArray(i, 3);
+                let expected = WinCondition_Reference(rst, abc);
+                let actual = Kata.WinCondition(rst, abc);
+
+                if actual != expected {
+                    Message($"Win condition '{actual}' is wrong for rst={rst}, abc={abc}");
+                    return false;
+                }
+            }
+        }
+        Message("Correct!");
+        true
+    }
+}

katas/content/nonlocal_games/ghz_win_condition/index.md

@@ -0,0 +1,9 @@
+**Inputs:**
+
+1. Alice, Bob and Charlie's input bits (r, s and t), stored as an array of length 3.
+   The input bits will have zero or two bits set to true.
+2. Alice, Bob and Charlie's output bits (a, b and c), stored as an array of length 3.
+
+**Goal:**
+
+True if Alice, Bob and Charlie won the GHZ game, that is, if R $\lor$ S $\lor$ T = A $\oplus$ B $\oplus$ C, and false otherwise.

katas/content/nonlocal_games/ghz_win_condition/solution.md

@@ -0,0 +1,11 @@
+There are four inputs possible, (0,0,0), (0,1,1), (1,0,1), and (1,1,0), each with $25\%$ probability.
+Therefore, in order to win, the sum of the output bits has to be even if the input is (0,0,0) and odd otherwise.
+
+To check whether the win condition holds, you need to compute the expressions R $\lor$ S $\lor$ T and A $\oplus$ B $\oplus$ C and to compare them:
+if they are equal, the game is won. To compute the expressions, you can use [built-in operators](https://learn.microsoft.com/azure/quantum/user-guide/language/expressions/logicalexpressions):
+X $\lor$ Y as `x or y` and A $\oplus$ B as `a != b`.
+
+@[solution]({
+    "id": "nonlocal_games__ghz_win_condition_solution",
+    "codePath": "Solution.qs"
+})

katas/content/nonlocal_games/index.md

@@ -31,10 +31,10 @@ In **CHSH Game**, two players (Alice and Bob) try to win the following game:
 
 Each of them is given a bit (Alice gets X and Bob gets Y), and
 they have to return new bits (Alice returns A and Bob returns B)
-so that X ∧ Y = A ⊕ B. The trick is, they can not communicate during the game.
+so that X $\land$ Y = A $\oplus$ B. The trick is, they can not communicate during the game.
 
-> - ∧ is the standard bitwise AND operator.
-> - ⊕ is the exclusive or, or XOR operator, so (P ⊕ Q) is true if exactly one of P and Q is true.
+> - $\land$ is the standard bitwise AND operator.
+> - $\oplus$ is the exclusive or, or XOR operator, so (P $\oplus$ Q) is true if exactly one of P and Q is true.
 
 To start with, let's take a look at how you would play the classical variant of this game without access to any quantum tools.
 Then, let's proceed with quantum strategies for Alice and Bob.
@@ -189,6 +189,43 @@ In the example below you can compare winning percentage of classical and quantum
 
 @[example]({"id": "nonlocal_games__chsh_e2edemo", "codePath": "./examples/CHSHGameDemo.qs"})
 
+@[section]({
+    "id": "nonlocal_games__ghz_game",
+    "title": "GHZ Game"
+})
+
+In **GHZ Game** three players (Alice, Bob and Charlie) try to win the following game:
+
+Each of them is given a bit (R, S and T respectively), and they have to return new bits (A, B and C respectively) so that 
+R $\lor$ S $\lor$ T = A $\oplus$ B $\oplus$ C.
+The input bits will have zero or two bits set to true and three or one bits set to false. 
+The players are free to share information (and even qubits!) before the game starts, but are forbidden from communicating
+with each other afterwards.
+
+> - $\lor$ is the standard bitwise OR operator.
+> - $\oplus$ is the exclusive or, or XOR operator, so (P $\oplus$ Q) is true if exactly one of P and Q is true.
+
+To start with, take a look at how you would play the classical variant of this game without access to any quantum tools.
+Then, let's proceed with quantum strategy and game implementation.
+
+@[exercise]({
+    "id": "nonlocal_games__ghz_win_condition",
+    "title": "Win Condition",
+    "path": "./ghz_win_condition/"
+})
+
+@[exercise]({
+    "id": "nonlocal_games__ghz_classical_strategy",
+    "title": "Classical Strategy",
+    "path": "./ghz_classical_strategy/"
+})
+
+@[exercise]({
+    "id": "nonlocal_games__ghz_classical_game",
+    "title": "Classical GHZ Game",
+    "path": "./ghz_classical_game/"
+})
+
 @[section]({
     "id": "nonlocal_games__conclusion",
     "title": "Conclusion"