Poker has become a popular pastime all over the world. At any given moment one can find tens, if not hundreds, of thousands of players playing poker via their computers on the major on-line gaming sites. Indeed, according to the Vancouver, B.C. based pokerpulse.com estimates, more than 190 million US dollars daily is bet on-line poker rooms. But communication and computation are changing as the relentless application of Moore's Law brings computation and information into the quantum realm. The quantum theory of games concerns the behavior of classical games when played the coming quantum computing environment or when played with quantum information. In almost all cases, the versions of these games afford many new strategic options to the players. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published Physics Review Letters 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. With the settling of some of these controversies \cite{Bleiler}, it is now possible to fully analyze some of the basic endgame models from the game theory of Poker and predict with confidence just how the optimal play of Poker will change when played the coming quantum computation environment. The analysis here shows that for certain players, entangled poker will allow results that outperform those available to players in real life.Poker has become a popular pastime all over the world. At any given moment one can find tens, if not hundreds, of thousands of players playing poker via their computers on the major on-line gaming sites. Indeed, according to the Vancouver, B.C. based pokerpulse.com estimates, more than 190 million US dollars daily is bet on-line poker rooms. But communication and computation are changing as the relentless application of Moore's Law brings computation and information into the quantum realm. The quantum theory of games concerns the behavior of classical games when played the coming quantum computing environment or when played with quantum information. In almost all cases, the versions of these games afford many new strategic options to the players. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published Physics Review Letters 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. With the settling of some of these controversies \cite{Bleiler}, it is now possible to fully analyze some of the basic endgame models from the game theory of Poker and predict with confidence just how the optimal play of Poker will change when played the coming quantum computation environment. The analysis here shows that for certain players, entangled poker will allow results that outperform those available to players in real life.

Feb 12, 20092 citations

A Formalism for Quantum Games and an Application

Steven A. Bleiler

This paper presents a new mathematical formalism that describes the quantization of games. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published in Physics Review Letters in 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. What has clouded many of the issues is the lack of a mathematical formalism for the subject in which these various issues can be clearly and precisely expressed, and which provides a context in which to present their resolution. Such a formalism is presented here, along with proposed resolutions to some of the issues discussed in the literature. One in particular, the question of whether there can exist equilibria in a quantized version of a game that do not correspond to classical correlated equilibria of that game and also deliver better payoffs than the classical correlated equilibria is answered in the affirmative for the Prisoner's Dilemma and Simplified Poker.This paper presents a new mathematical formalism that describes the quantization of games. The study of so-called quantum games is quite new, arising from a seminal paper of D. Meyer \cite{Meyer} published in Physics Review Letters in 1999. The ensuing near decade has seen an explosion of contributions and controversy over what exactly a quantized game really is and if there is indeed anything new for game theory. What has clouded many of the issues is the lack of a mathematical formalism for the subject in which these various issues can be clearly and precisely expressed, and which provides a context in which to present their resolution. Such a formalism is presented here, along with proposed resolutions to some of the issues discussed in the literature. One in particular, the question of whether there can exist equilibria in a quantized version of a game that do not correspond to classical correlated equilibria of that game and also deliver better payoffs than the classical correlated equilibria is answered in the affirmative for the Prisoner's Dilemma and Simplified Poker.

arXiv:0808.1389Aug 10, 200823 citations

Quantization of blackjack: Quantum basic strategy and advantage

Yushi Mura, Hiroki Wada

Abstract Quantum computers that process information by harnessing the remarkable power of quantum mechanics are increasingly being put to practical use. In the future, their impact will be felt in numerous fields, including in online casino games. This is one of the reasons why quantum gambling theory has garnered considerable attention. Studies have shown that quantum gambling theory often yields nontrivial consequences that classical theory cannot interpret. We have formulated blackjack, one of the most famous card games, as a quantum game and found possible quantum entanglement between strategies. We also devised a quantum circuit reproducing classical blackjack. This circuit can be realized in the near future when quantum computers are commonplace. Furthermore, we showed that the player’s expectation increases compared to the classical game using quantum basic strategy, which is a quantum version of the popular basic strategy of blackjack.Abstract Quantum computers that process information by harnessing the remarkable power of quantum mechanics are increasingly being put to practical use. In the future, their impact will be felt in numerous fields, including in online casino games. This is one of the reasons why quantum gambling theory has garnered considerable attention. Studies have shown that quantum gambling theory often yields nontrivial consequences that classical theory cannot interpret. We have formulated blackjack, one of the most famous card games, as a quantum game and found possible quantum entanglement between strategies. We also devised a quantum circuit reproducing classical blackjack. This circuit can be realized in the near future when quantum computers are commonplace. Furthermore, we showed that the player’s expectation increases compared to the classical game using quantum basic strategy, which is a quantum version of the popular basic strategy of blackjack.

arXiv:2011.12342Progress of Theoretical and Experimental PhysicsSep 29, 20212 citations

Quantum Poker—a game for quantum computers suitable for benchmarking error mitigation techniques on NISQ devices

Franz Georg Fuchs, Vemund Falch, Christian Johnsen

Abstract Quantum computers are on the verge of becoming a commercially available reality. They represent a paradigm shift in computing, with a steep learning gradient. The creation of games is a way to ease the transition for beginners. We present a game similar to the Poker variant Texas hold ’em with the intention to serve as an engaging pedagogical tool to learn the basics rules of quantum computing. The concepts of quantum states, quantum operations and measurement can be learned in a playful manner. The difference to the classical variant is that the community cards are replaced by a quantum register that is “randomly” initialized, and the cards for each player are replaced by quantum gates, randomly drawn from a set of available gates. Each player can create a quantum circuit with their cards, with the aim to maximize the number of 1’s that are measured in the computational basis. The basic concepts of superposition, entanglement and quantum gates are employed. We provide a proof-of-concept implementation using Qiskit (Aleksandrowicz et al. in An open-source framework for quantum computing, 2019). A comparison of the results for the created circuits using a simulator and IBM machines is conducted, showing that error rates on contemporary quantum computers are still very high. For the success of noisy intermediate scale quantum (NISQ) computers, improvements on the error rates and error mitigation techniques are necessary, even for simple circuits. We show that quantum error mitigation (QEM) techniques can be used to improve expectation values of observables on real quantum devices.Abstract Quantum computers are on the verge of becoming a commercially available reality. They represent a paradigm shift in computing, with a steep learning gradient. The creation of games is a way to ease the transition for beginners. We present a game similar to the Poker variant Texas hold ’em with the intention to serve as an engaging pedagogical tool to learn the basics rules of quantum computing. The concepts of quantum states, quantum operations and measurement can be learned in a playful manner. The difference to the classical variant is that the community cards are replaced by a quantum register that is “randomly” initialized, and the cards for each player are replaced by quantum gates, randomly drawn from a set of available gates. Each player can create a quantum circuit with their cards, with the aim to maximize the number of 1’s that are measured in the computational basis. The basic concepts of superposition, entanglement and quantum gates are employed. We provide a proof-of-concept implementation using Qiskit (Aleksandrowicz et al. in An open-source framework for quantum computing, 2019). A comparison of the results for the created circuits using a simulator and IBM machines is conducted, showing that error rates on contemporary quantum computers are still very high. For the success of noisy intermediate scale quantum (NISQ) computers, improvements on the error rates and error mitigation techniques are necessary, even for simple circuits. We show that quantum error mitigation (QEM) techniques can be used to improve expectation values of observables on real quantum devices.

arXiv:1908.00044The European Physical Journal PlusApr 1, 202034 citations

An Investigation of Quantum Computational Game Theory

Scott Ricketts, Max Mintz

Game theory is a well established field with applications in many areas including the social sciences, biology, and computer science [6]. Recently, however, researchers in quantum computation have begun to study the implications of games set in the quantum domain (i.e. game theoretical situations where either one or more players have quantum computational power or where the implementation of strategies uses some form of quantum information). Such a setting allows for entanglement and superpositions of strategies and evaluation of payoffs using quantum measurement. This approach to game theory, according to some scholars in the field, could help us understand certain physical systems, improve economic markets, and develop quantum algorithms. I have conducted an investigation into quantum game theory. I have constructed simulation tools using Maple that, along with research into related work, have helped gain some insight into this new field. The important solutions in classical game theory are usually Nash equilibria. What do they look like in the quantum realm? In whatGame theory is a well established field with applications in many areas including the social sciences, biology, and computer science [6]. Recently, however, researchers in quantum computation have begun to study the implications of games set in the quantum domain (i.e. game theoretical situations where either one or more players have quantum computational power or where the implementation of strategies uses some form of quantum information). Such a setting allows for entanglement and superpositions of strategies and evaluation of payoffs using quantum measurement. This approach to game theory, according to some scholars in the field, could help us understand certain physical systems, improve economic markets, and develop quantum algorithms. I have conducted an investigation into quantum game theory. I have constructed simulation tools using Maple that, along with research into related work, have helped gain some insight into this new field. The important solutions in classical game theory are usually Nash equilibria. What do they look like in the quantum realm? In what

Jan 1, 20060 citations

Quantumizing Classical Games: An Introduction to Quantum Game Theory

Sowmitra Das

We give a concise and self-contained introduction to the theory of Quantum Games by reviewing the seminal works of Meyer, Eisert-Wilkens-Lewenstein, Marinatto-Weber and Landsburg, which initiated the study of this field. By generalizing this body of work, we formulate a protocol to $\textit{Quantumize}$ any finite classical $n$ -player game, and use a novel approach of describing such a Quantum Game in terms of commuting Payoff Operators. We describe what advantages can be gained by players by quantumizing such a game, particularly, what additional Nash Equilibria the players can achieve and the Pareto-Optimality of these additional equilibria.We give a concise and self-contained introduction to the theory of Quantum Games by reviewing the seminal works of Meyer, Eisert-Wilkens-Lewenstein, Marinatto-Weber and Landsburg, which initiated the study of this field. By generalizing this body of work, we formulate a protocol to $\textit{Quantumize}$ any finite classical $n$-player game, and use a novel approach of describing such a Quantum Game in terms of commuting Payoff Operators. We describe what advantages can be gained by players by quantumizing such a game, particularly, what additional Nash Equilibria the players can achieve and the Pareto-Optimality of these additional equilibria.

arXiv:2305.00368Apr 30, 20230 citations

\textit{Quantumize}

any finite classical

n

-player game, and use a novel approach of describing such a Quantum Game in terms of commuting Payoff Operators. We describe what advantages can be gained by players by quantumizing such a game, particularly, what additional Nash Equilibria the players can achieve and the Pareto-Optimality of these additional equilibria.