Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2104.12043 · Mathematics

The Connection between the PQ Penny Flip Game and the Dihedral Groups

Theodore Andronikos, Alla Sirokofskich

Published May 14, 2021• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.3390/math9101115

Related papers

Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

Theodore Andronikos, Alla Sirokofskich, Kalliopi Kastampolidou +3 more

The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The PQ penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper we investigate all possible finite games that can be played between the two players Q and Picard of the original PQ game. For this purpose we establish a rigorous connection between finite automata and the PQ game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the PQ game. What this means is that from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player.

Mathematics•Feb 1, 2018•29 citations

Non-Abelian strategies in quantum penny flip game

Hiroaki Mishima

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.

arXiv:1707.00938•Progress of Theoretical and Experimental Physics•Nov 29, 2017•1 citation

Arbitrary Finite Dimensional Generalization of Quantum Coin-Flipping Game

Fasheng Cao

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.

Journal of Physics Conference Series•Oct 1, 2020•0 citations

An Analysis of the Quantum Penny Flip Game using Geometric Algebra

James M. Chappell, Azhar Iqbal, Max A. Lohe +1 more

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.

arXiv:0902.4296•Journal of the Physical Society of Japan•May 11, 2009•23 citations

Fair Game of Quantum Penny Flip

Xiaoken Yang, Wei Li, Yong-Chang Huang

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.

Advances in transdisciplinary engineering•Oct 17, 2023•0 citations

Conditions that Enable a Player to Surely Win in Sequential Quantum Games

Theodore Andronikos

This paper studies sequential quantum games under the assumption that the moves of the players are drawn from groups and not just plain sets. The extra group structure makes possible to easily derive some very general results, which, to the best of our knowledge, are stated in this generality for the first time in the literature. The main conclusion of this paper is that the specific rules of a game are absolutely critical. The slightest variation may have important impact on the outcome of the game. It is the combination of two factors that determine who wins: (i) the sets of admissible moves for each player, and (ii) the order of moves, i.e., whether the same player makes the first and the last move. Quantum strategies do not a priori prevail over classical strategies. By carefully designing the rules of the game the advantage of either player can be established. Alternatively, the fairness of the game can also be guaranteed.

•Apr 12, 2021•1 citation
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.
The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The PQ penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper we investigate all possible finite games that can be played between the two players Q and Picard of the original PQ game. For this purpose we establish a rigorous connection between finite automata and the PQ game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the PQ game. What this means is that from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player.
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)U(2)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.
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.
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.
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.
This paper studies sequential quantum games under the assumption that the moves of the players are drawn from groups and not just plain sets. The extra group structure makes possible to easily derive some very general results, which, to the best of our knowledge, are stated in this generality for the first time in the literature. The main conclusion of this paper is that the specific rules of a game are absolutely critical. The slightest variation may have important impact on the outcome of the game. It is the combination of two factors that determine who wins: (i) the sets of admissible moves for each player, and (ii) the order of moves, i.e., whether the same player makes the first and the last move. Quantum strategies do not a priori prevail over classical strategies. By carefully designing the rules of the game the advantage of either player can be established. Alternatively, the fairness of the game can also be guaranteed.