We analyze the quantum penny flip game using geometric algebra and so determine all possible unitary transformations which enable the player Q to implement a winning strategy. Geometric algebra provides a clear visual picture of the quantum game and its strategies, as well as providing a simple and direct derivation of the winning transformation, which we demonstrate can be parametrized by two angles. For comparison we derive the same general winning strategy by conventional means using density matrices.
We analyze the quantum penny flip game using geometric algebra and so determine all possible unitary transformations which enable the player Q to implement a winning strategy. Geometric algebra provides a clear visual picture of the quantum game and its strategies, as well as providing a simple and direct derivation of the winning transformation, which we demonstrate can be parametrized by two angles. For comparison we derive the same general winning strategy by conventional means using density matrices.
In this paper, we formulate and analyze generalizations of the quantum penny flip game. In the penny flip game, one coin has two states, heads or tails, and two players apply alternating operations on the coin. In the original Meyer game, the first player is allowed to use quantum (i.e., non-commutative) operations, but the second player is still only allowed to use classical (i.e., commutative) operations. In our generalized games, both players are allowed to use non-commutative operations, with the second player being partially restricted in what operators they use. We show that even if the second player is allowed to use “phase-variable” operations, which are non-Abelian in general, the first player still has winning strategies. Furthermore, we show that even when the second player is allowed to choose one from two or more elements of the group |U(2)|, the second player has winning strategies under certain conditions. These results suggest that there is often a method for restoring the quantum state disturbed by another agent.
This paper is inspired by the PQ penny flip game. It employs group-theoretic concepts to study the original game and its possible extensions. In this paper, it is shown that the PQ penny flip game can be associated, in a precise way, with the dihedral group D8 and that within D8 there exist precisely two classes of equivalent winning strategies for Q. This is achieved by proving that there are exactly two different sequences of states that can guarantee Q’s win with probability 1.0. It is demonstrated that the game can be played in every dihedral group D8n, where n≥1, without any significant change. A formal examination of what happens when Q can draw their moves from the entire U(2), leads to the conclusion that, again, there are exactly two classes of winning strategies for Q, each class containing an infinite number of equivalent strategies, but all of them sending the coin through the same sequence of states as before. Finally, when general extensions of the game, with the quantum player having U(2) at their disposal, are considered, a necessary and sufficient condition for Q to surely win against Picard is established: Q must make both the first and the last move in the game.
Abstract Two players of coin-flipping game randomly choose one of two strategies, flipping over or not. Meyer quantized states of the game to be described instead by density matrices of the two-dimensional Hilbert space. In order to obtain and unify generation of the two players coin-flipping game with more than two strategies, we renew the need for equivalently describing Meyer’s PQ coin-flipping game by representing player’s strategies as elements of permutation groups. Expanding the dimension of quantum vector to arbitrary N-dimension in the coin-flipping game model, we generalize Meyer’s PQ coin-flipping game to PQ arbitrary finite strategies game. Owing to the quantum Fourier transform, quantum contrivance of Q has a winning strategy in PQ coin-flipping game with arbitrary finite strategies. Even though the player Q has a small probability of winning his opponent in the classical PQ coin-flipping game with arbitrary finite strategies, he can win his opponent who uses classical mixed strategies by his quantum strategies. Furthermore, we realize the winning quantum strategies of quantum player Q of PQ coin-flipping game with arbitrary finite strategies in macro world.
The Quantum Penny Flip is a classic thought experiment in quantum mechanics that demonstrates the weird and counterintuitive nature of the quantum world. To solve the “PQ” problem, we proposed a quantum strategy selection and optimization algorithm, and designed a quantum algorithm for this problem. In terms of practical applications and Case study, this paper reviews the classic two player game – “PQ coin flipping” problem, and shows that when one of the participants adopts quantum strategies, he can defeat his classic opponent and obtain higher returns. By using quantitative methods, the entire “PQ” problem is placed under a more general and fair condition, and it is promoted and studied again from multiple aspects and perspectives. At the same time, this article provides the definitions of “incomplete fair game” and “complete fair game”, modifies the quantum coin flipping game [Meyer, Phys. Rev. lett. 82 (1999) 1052] to make the game fair, and studies the multi round versions of quantum coin flipping. We have drawn some basic conclusions about fair quantum games and explained the meaning of completely fair games in practice. We discussed the limitations and scalability of the “PQ” problem and looked forward to the future development direction of quantum game research.
The framework for playing quantum games in an Einstein-Podolsky-Rosen (EPR) type setting is investigated using the mathematical formalism of geometric algebra (GA). The main advantage of this framework is that the players' strategy sets remain identical to the ones in the classical mixed-strategy version of the game, and hence the quantum game becomes a proper extension of the classical game, avoiding a criticism of other quantum game frameworks. We produce a general solution for two-player games, and as examples, we analyze the games of Prisoners' Dilemma and Stag Hunt in the EPR setting. The use of GA allows a quantum-mechanical analysis without the use of complex numbers or the Dirac Bra-ket notation, and hence is more accessible to the non-physicist.
The high public attention given to quantum computing shows that it is perceived as an interesting topic. We want to utilize this motivating effect for the teaching and learning of quantum physics. Specifically, we want to take advantage of the access to real quantum computers, which various providers make available free of charge. We show a direct and straightforward way to use quantum computers in an introductory course on quantum physics. We use an algorithm that solves a simple and easily understandable problem while providing a quantum advantage. The algorithm we propose is a simple game in which the use of quantum physics offers a winning advantage. The game is called Quantum Penny Flip and was proposed by David A. Meyer back in 1999. It can be easily reformulated to be described by quantum gates. We can therefore use it to teach the programming of a quantum computer. We demonstrate its implementation in IBM's Quantum Composer.