# CHSH Game¶

In this example, we show how to play the CHSH game with QuNetSim. The CHSH game is an example where a quantum strategy outperforms or wins more often, than any classical strategy. In the game, there are three parties. A referee and two cooperating players, Alice and Bob. The point of the game is, the referee sends Alice and Bob each a random bit, $$x$$ to Alice and $$y$$ to Bob, that is, with probability $$0.5$$ Alice and Bob will receive a $$0$$ and with $$0.5$$ a $$1$$. During the game, Alice and Bob cannot communicate with each other, but to win the game they each have to send a bit back to the referee such that the equation $$x$$ and $$y = a$$ xor $$b$$, where $$a$$ and $$b$$ are the bits Alice and Bob send to the referee.

Before the game, Alice and Bob can devise a strategy and distribute resources that they can use during the game, as long as the resource is not a method of communication. The optical strategy for winning the game classically is that no matter what the referee sends them, Alice and Bob both send a $$0$$. With this strategy, they can win $$75\%$$ of the time. If they instead use a quantum strategy, before the game starts, they can distribute amongst themselves entangled pairs of qubits. Then, depending on the bit that the referee sends then, they perform a specific measurement on their half of the qubits. With this strategy, they can win roughly $$85\%$$ of the time.

Below, we play the game both classically and quantumly.

The protocol that the referee runs the following. The referee simply generates 2 random bits and sends them to Alice and Bob respectively. He then awaits a response from both Alice and Bob and then records the statistics for if Alice and Bob won or lost.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 def referee(ref, alice_id, bob_id): """ Referee protocol for CHSH game. Args: ref (Host): Referee host object alice_id (str): Alice's host ID bob_id (str): Bob's host ID """ wins = 0 for i in range(PLAYS): x = random.choice([0, 1]) ref.send_classical(alice_id, str(x)) y = random.choice([0, 1]) ref.send_classical(bob_id, str(y)) alice_response = ref.get_classical(alice_id, seq_num=i, wait=5) bob_response = ref.get_classical(bob_id, seq_num=i, wait=5) a = int(alice_response.content) b = int(bob_response.content) print('X, Y, A, B --- %d, %d, %d, %d' % (x, y, a, b)) if x & y == a ^ b: print('Winners!') wins += 1 else: print('Losers!') print("Win ratio: " + "{0:.2%}".format(1. * wins / PLAYS)) 

Next we see Alice’s protocols for the classical and quantum strategies. In the classical strategy, she simply awaits a message from the referee and then sends back a 0 in all cases.

Quantumly, when she receives the message from the referee, she performs either a $$Z$$ basis measurement or and $$X$$ basis.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34  def alice_classical(alice_host, referee_id): """ Alice's classical protocol for the CHSH game. Args: alice_host (Host): Alice's Host object referee_id (str): Referee's Host ID """ for i in range(PLAYS): _ = alice_host.get_classical(referee_id, seq_num=i, wait=5) alice_host.send_classical(referee_id, "0") def alice_quantum(alice_host, referee_id, bob_id): """ Alice's quantum protocol for the CHSH game. Args: alice_host (Host): Alice's Host object referee_id (str): Referee's Host ID bob_id (str): Bob's Host ID (only for accessing shared EPR pairs) """ for i in range(PLAYS): referee_message = alice_host.get_classical(referee_id, seq_num=i, wait=5) x = int(referee_message.content) epr = alice_host.get_epr(bob_id) if x == 0: res = epr.measure() alice_host.send_classical(referee_id, str(res)) else: epr.H() res = epr.measure() alice_host.send_classical(referee_id, str(res)) 

Classically, Bob does uses the exact same strategy. Quantumly Bob has a similar approach except that he uses another basis for his measurements, namely a rotated basis in the Y-axis.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 def bob_classical(bob_host, referee_id): """ Bob's classical protocol for the CHSH game. Args: bob_host (Host): Bob's Host object referee_id (str): Referee's Host ID """ for i in range(PLAYS): _ = bob_host.get_classical(referee_id, seq_num=i, wait=5) bob_host.send_classical(referee_id, "0") def bob_quantum(bob_host, referee_id, alice_id): """ Bob's quantum protocol for the CHSH game. Args: bob_host (Host): Bob's Host object referee_id (str): Referee's Host ID alice_id (str): Alice's Host ID (only for accessing shared EPR pairs) """ for i in range(PLAYS): referee_message = bob_host.get_classical(referee_id, seq_num=i, wait=5) y = int(referee_message.content) epr = bob_host.get_epr(alice_id) if y == 0: epr.ry(-2.0 * math.pi / 8.0) res = epr.measure() bob_host.send_classical(referee_id, str(res)) else: epr.ry(2.0 * math.pi / 8.0) res = epr.measure() bob_host.send_classical(referee_id, str(res)) 

For this example, because we are using rotation operators, we need to change the system backend to one that supports such operations. QuNetSim supports, at the moment, three backends for qubits, namely, SimulaQron, ProjectQ, and a qubit simulator that we’ve developed called ESQN. ProjectQ and EQSN both support rotation operators and so we use either of those here. The full example is below, with the backends imported and set.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 import math import random from qunetsim.components import Host from qunetsim.components import Network from qunetsim.objects import Logger # Disable QuNetSim logging Logger.DISABLED = True # Number of times to play the game. PLAYS = 20 # Classical or Quantum strategy for the game # strategy = 'CLASSICAL' strategy = 'QUANTUM' def alice_classical(alice_host, referee_id): """ Alice's classical protocol for the CHSH game. Args: alice_host (Host): Alice's Host object referee_id (str): Referee's Host ID """ for i in range(PLAYS): _ = alice_host.get_classical(referee_id, seq_num=i, wait=5) alice_host.send_classical(referee_id, "0") def alice_quantum(alice_host, referee_id, bob_id): """ Alice's quantum protocol for the CHSH game. Args: alice_host (Host): Alice's Host object referee_id (str): Referee's Host ID bob_id (str): Bob's Host ID (only for accessing shared EPR pairs) """ for i in range(PLAYS): referee_message = alice_host.get_classical(referee_id, seq_num=i, wait=5) x = int(referee_message.content) epr = alice_host.get_epr(bob_id) if x == 0: res = epr.measure() alice_host.send_classical(referee_id, str(res)) else: epr.H() res = epr.measure() alice_host.send_classical(referee_id, str(res)) def bob_classical(bob_host, referee_id): """ Bob's classical protocol for the CHSH game. Args: bob_host (Host): Bob's Host object referee_id (str): Referee's Host ID """ for i in range(PLAYS): _ = bob_host.get_classical(referee_id, seq_num=i, wait=5) bob_host.send_classical(referee_id, "0") def bob_quantum(bob_host, referee_id, alice_id): """ Bob's quantum protocol for the CHSH game. Args: bob_host (Host): Bob's Host object referee_id (str): Referee's Host ID alice_id (str): Alice's Host ID (only for accessing shared EPR pairs) """ for i in range(PLAYS): referee_message = bob_host.get_classical(referee_id, seq_num=i, wait=5) y = int(referee_message.content) epr = bob_host.get_epr(alice_id) if y == 0: epr.ry(-2.0 * math.pi / 8.0) res = epr.measure() bob_host.send_classical(referee_id, str(res)) else: epr.ry(2.0 * math.pi / 8.0) res = epr.measure() bob_host.send_classical(referee_id, str(res)) def referee(ref, alice_id, bob_id): """ Referee protocol for CHSH game. Args: ref (Host): Referee host object alice_id (str): Alice's host ID bob_id (str): Bob's host ID """ wins = 0 for i in range(PLAYS): x = random.choice([0, 1]) ref.send_classical(alice_id, str(x)) y = random.choice([0, 1]) ref.send_classical(bob_id, str(y)) alice_response = ref.get_classical(alice_id, seq_num=i, wait=5) bob_response = ref.get_classical(bob_id, seq_num=i, wait=5) a = int(alice_response.content) b = int(bob_response.content) print('X, Y, A, B --- %d, %d, %d, %d' % (x, y, a, b)) if x & y == a ^ b: print('Winners!') wins += 1 else: print('Losers!') print("Win ratio: %.2f" % (100. * wins / PLAYS)) def main(): network = Network.get_instance() network.start() host_A = Host('A') host_A.add_c_connection('C') host_A.start() host_B = Host('B') host_B.add_c_connection('C') host_B.start() host_C = Host('C') host_C.add_c_connection('A') host_C.add_c_connection('B') host_C.start() network.add_host(host_C) # For entanglement generation host_A.add_connection('B') host_B.add_connection('A') network.add_host(host_A) network.add_host(host_B) if strategy == 'QUANTUM': print('Generating initial entanglement') for i in range(PLAYS): host_A.send_epr('B', await_ack=True) print('Done generating initial entanglement') # Remove the connection from Alice and Bob host_A.remove_connection('B') host_B.remove_connection('A') network.update_host(host_A) network.update_host(host_B) print('Starting game. Strategy: %s' % strategy) # Play the game classically if strategy == 'CLASSICAL': host_A.run_protocol(alice_classical, (host_C.host_id,)) host_B.run_protocol(bob_classical, (host_C.host_id,), ) # Play the game quantumly if strategy == 'QUANTUM': host_A.run_protocol(alice_quantum, (host_C.host_id, host_B.host_id)) host_B.run_protocol(bob_quantum, (host_C.host_id, host_A.host_id)) host_C.run_protocol(referee, (host_A.host_id, host_B.host_id), blocking=True) if __name__ == '__main__': main()