Running Picard and Q Game on IBM's Quantum Computers

Quantum game theory has evolved through the application of quantum methods to game theory scenarios and it has been used in two settings:

• To simulate human decision making, in which there has been evidence of quantum-like interference patterns when humans evaluate different alternatives;
• To simulate the  context where players have access to quantum computers and can play with quantum strategies.

These two settings have led to the development of quantum econophysics with the simulation of financial markets being one of the major contributions (Piotrowski and Sładkowski, 2017; Gonçalves, 2013, 2015).

In the current blog post, we implement an early example of quantum strategies beating classical strategies, using the cloud-based access to IBM's Quantum Computers, in the next blog post we will expand the results to a quantum cybersecurity game.

To build the illustration game for the current blog post, we return to David Meyer's (1999) original article which introduced quantum strategies through the Picard and Q game. 

For the present purposes, let us consider first the classical version of the game:

Picard and Q are playing a game where they either turn a coin or not without looking at it, Q goes first, then Picard, then Q goes again, they only look at the result in the end, if the coin ends up heads then Picard wins, if it ends up tails then Q wins. The coin is initially with tails facing up.

Each player is playing without knowing what the other player did, so that we have the following scenarios:

1. Q does not turn the coin -> Picard does not turn the coin -> Q does not turn the coin. In this case, the coin sequence from its initial position is:

Tails->Tails->Tails->Tails: Q wins (+1), Picard loses (-1)

2. Q does not turn the coin->Picard does not turn the coin -> Q turns the coin. In this case, the coin sequence from its initial position is:

Tails->Tails->Tails->Heads: Q loses (-1), Picard wins (+1)

3. Q turns the coin -> Picard does not turn the coin -> Q does not turn the coin. In this case, the coin sequence from its initial position is:

Tails->Heads->Heads->Heads: Q loses (-1), Picard wins (+1) 

4. Q turns the coin -> Picard does not turn the coin -> Q turns the coin. In this case, the coin sequence from its initial position is:

Tails->Heads->Heads->Tails: Q wins (+1), Picard loses (-1)

5. Q does not turn the coin -> Picard turns the coin -> Q does not turn the coin. In this case, the coin sequence from its initial position is: 

Tails->Tails->Heads->Heads: Q loses (-1), Picard wins (+1)

6. Q does not turn the coin -> Picard turns the coin -> Q turns the coin. In this case, the coin sequence from its initial position is: 

Tails->Tails->Heads->Tails: Q wins (+1), Picard loses (-1)

7. Q turns the coin -> Picard turns the coin -> Q does not turn the coin. In this case, the coin sequence from its initial position is:

Tails->Heads->Tails->Tails: Q wins (+1), Picard loses (-1) 

8. Q turns the coin -> Picard turns the coin -> Q turns the coin. In this case, the coin sequence from its initial position is:

Tails->Heads->Tails->Heads: Q loses (-1), Picard wins (+1)

These eight scenarios are summarized in the following game matrix:

Game	Picard
Q	Not Turn	Turn
Not Turn, Not Turn	+1,-1	-1,+1
Not Turn, Turn	-1,+1	+1,-1
Turn, Not Turn	-1,+1	+1,-1
Turn, Turn	+1,-1	-1,+1

There are no pure strategies Nash equilibria for this game, however, From Nash's original work (Nash, 1950) we know that such finite games always have an equilibrium, which means that if it is not a pure strategies equilibrium, then it is a mixed strategies equilibrium.

If we consider the classical probabilities we find that if Picard plays Not Turn with 50% probability and Turn with 50% probability then Q is indifferent between each classical pure strategy, that is, he does not win more by changing to one pure strategy in detriment of the other.

On the other hand, if Q plays with 50% probability the solution Turn, Turn or Not Turn, Not Turn, and with 50% probability the solution Turn, Not Turn or Not Turn, Turn, then, Picard will be indifferent between playing any one of his pure strategies that is, he does not win more by changing to one pure strategy in detriment of another.

However, if Q plays a specific quantum strategy, then Q can win every time! That is, there is a quantum strategy that is dominant for Q.

Indeed, let us replace the classical coin by a quantum register representing a quantum coin, with |0> encoding tails and |1> encoding heads, then Q may apply the Haddamard transform leading to the symmetric superposition:

|+> = (|0> + |1>)/sqrt(2) 

Now, if Picard applies a (classical) pure strategy, then, he can either apply the identity gate (not turn the quantum coin) or the NOT gate (turn the quantum coin).

Let us assume that after Picard plays his pure strategy, Q again applies a Haddamard transform, so that we have the following two quantum circuits for each alternative sequence of plays:

Picard does not turn the quantum coin:

Picard turns the quantum coin:

Let us, now, see what happens in each case. If we run each circuit on the IBM's device ibmq_qasm_simulator, then, for the first circuit, when we finally measure the result from playing the game, we get, in 1024 repeated experiments, the following results:

That is, the game always leads to the logical state 0 (Tails), which means that Q always wins.

And what about the second circuit?

If we run it we get, in 1024 repeated experiments, the following results:

Again, Q always wins. So Q always wins independently of what Picard does! Why is that?

Well, the answer is symmetry! Q has placed the register in a symmetric superposition of |0> and |1>, which means that when Picard turns the quantum coin the superposition is unchanged, the same is true when Picard does not turn the quantum coin, the result it is invariant with respect to whatever Picard does, after Picard's move the quantum coin is always in the superposition |+>, which means that Q can now replay his move applying the Haddamard gate which inverts his initial operation and returns it to |0>, that is, to Tails, and so he wins every time.

This also means that if Picard turns with probability "p" or not with probability "1-p", the result is always the same, Q always wins, whatever the strategy, pure or mixed, played by Picard.

This shows how quantum game theory can go beyond the classical results and, playing with quantum strategies, a player can win over the classical Nash equilibrium. In this case, the quantum strategy is strategically dominant for Q over the classical mixed strategy, that is, he always wins more than he would by playing the classical mixed strategy. Meyer (1999) effectively expanded the issue of strategic dominance from the classical level to the quantum level.

Now, we have so far run the game on the simulator, what if we run the game on an actual quantum computer? If we run it on the device ibmq_london, for the first circuit, using 1024 repeated experiments, we get the following results:

As we can see, in 99.707% of the time, Q wins, in the remaining cases, Q loses. It is no longer absolutely certain that Q may win, why is that? 

The answer is that when we ran the circuit in the simulator we were working with an isolated quantum computer (the theoretical device), which means that there was no error, when we run the circuit on the actual physical device, environmental fluctuations introduce error in the quantum computation, which means that there is always risk present, so that Q and Picard are playing a game with each other and with Nature. 

Therefore, Q wins... most of the time, but not always, however, above 90% probability of winning (in this case very near 100%) is good enough.

The same holds for the second circuit, using again 1024 repeated experiments and the same quantum device, we get:

That is, Q wins 92.383% of the time. So, again, Q wins more than 90% of the time, which is still an advantage over Picard.

The above results show how quantum strategies can be implemented on actual quantum computers, and how quantum strategies can win over classical ones, in the next blog post we will show an example that expands this post's illustration to the subject matter of quantum cybersecurity.

Consulted References

- Piotrowski, E.W., Sładkowski, J. (2017). Quantum game theoretical frameworks in economics. In Haven, Emmanuel (ed.) et al., The Palgrave handbook of quantum models in social science. Applications and grand challenges. New York, NY: Palgrave Macmillan.

- Gonçalves, C.P. (2015). Financial Market Modeling with Quantum Neural Networks. Review of Business and Economics Studies (ROBES), 3(4): 44-63.

- Gonçalves, C.P. (2013). Quantum financial economics - risk and returns. Journal of Systems Science and Complexity, 26(2): 187-200.

- Meyer, D.A. (1999). Quantum Strategies. Phys. Rev. Lett. 82, 1052.

- Nash, J. (1950). Non-Cooperative Games. PhD Thesis, Princeton University.