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

Advanced Quantum Technologies

Analysis of the Effect of Entanglement Operators and the Scalability of Players’ Payoff Computation in N‐Player Quantum Games

Georgios D. Varsamis, Ioannis Liliopoulos, Hamed Mohammadbagherpoor, Andreas K. Kostopoulos, Kristin L. Milchanowski, Evangelos T. Karamatskos, Panagiotis Dimitrakis, Richard P. Padbury, Ioannis G. Karafyllidis

Published Aug 18, 2025• 2 citations
Publisher

Abstract

Abstract Entanglement is one of the most vital resources in quantum computing. Particularly, in quantum game theory, entanglement holds significant importance since it defines the interactions and the information exchange between two or more players. In this research work, a new set of entanglement operators is introduced for N‐player quantum games. This study explores how their application affects each player's behavior throughout the game. Furthermore, the entangling capabilities are tested and evaluated for a wide range of entanglement operator angles, by exploiting the Von Neumann entropy metric, both on a simulator and on a real IBM Quantum system. Their hardware efficiency, against state‐of‐the‐art quantum games entanglement operators, is assessed in terms of circuit depths and fidelity, and showcases their great potential as multiplayer quantum games operators. Finally, three methods are compared for computing the players’ payoffs, regarding their scalability and efficiency. This study shows that in all payoff cases, the quantum game results, computed by the quantum hardware, are in accordance with the simulated ones.

DOI
10.1002/qute.202500375

Related papers

Constructing Entanglers in 2-Players–N-Strategies Quantum Game

Y. Avishai

In quantum games based on 2-player-N-strategies classical games, each player has a quNit (a normalized vector in an N-dimensional Hilbert space HN) upon which he applies his strategy (a matrix U∈SU(N)). The players draw their payoffs from a state . Here and J (both determined by the game’s referee) are respectively an unentangled 2-quNit (pure) state and a unitary operator such that is partially entangled. The existence of pure strategy Nash equilibrium in the quantum game is intimately related to the degree of entanglement of . Hence, it is practical to design the entangler J= J(β) to be dependent on a single real parameter β that controls the degree of entanglement of , such that its von-Neumann entropy SN(β) is continuous and obtains any value in . Designing J(β) for N=2 is quite standard. Extension to N>2 is not obvious, and here we suggest an algorithm to achieve it. Such construction provides a special quantum gate that should be a useful tool not only in quantum games but, more generally, as a special gate in manipulating quantum information protocols.

arXiv:1402.1982•Journal of Quantum Information Science•Jan 1, 2015•5 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 what

•Jan 1, 2006•0 citations

The Superiority of Quantum Strategy in 3-Player Prisoner’s Dilemma

Zhiyuan Dong, Ai‐Guo Wu

In this paper, we extend the quantum game theory of Prisoner’s Dilemma to the N-player case. The final state of quantum game theory of N-player Prisoner’s Dilemma is derived, which can be used to investigate the payoff of each player. As demonstration, two cases (2-player and 3-player) are studied to illustrate the superiority of quantum strategy in the game theory. Specifically, the non-unique entanglement parameter is found to maximize the total payoff, which oscillates periodically. Finally, the optimal strategic set is proved to depend on the selection of initial states.

Mathematics•Jun 21, 2021•1 citation

Quantum Game Simulator, Using the Circuit Model of Quantum

Panagiotis Vlachos, Ioannis G. Karafyllidis

We present a general two-player quantum game simulator that can simulate any two-player quantum game described by a 2times2 payoff matrix (two strategy games).The user can determine the payoff matrices for both players, their strategies and the amount of entanglement between their initial strategies. The outputs of the simulator are the expected payoffs of each player as a function of the other player's strategy parameters and the amount of entanglement. The simulator also produces contour plots that divide the strategy spaces of the game in regions in which players can get larger payoffs if they choose to use a quantum strategy against any classical one. We also apply the simulator to two well known quantum games, the Battle of Sexes and the Chicken game.

Panhellenic Conference on Informatics•Jan 1, 2009•1 citation

A necessary and sufficient condition to play games in quantum mechanical settings

Şahin Kaya Özdemir, Junichi Shimamura, N. Imoto

Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player two-strategy (2 × 2) dilemma containing classical games into the quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2 × 2 games, it became generally accepted that quantum versions can be easily extended to N -player situations by simply allowing N -partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in the quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.

arXiv:quant-ph/0703006•New Journal of Physics•Feb 28, 2007•57 citations

Proxy Quantum Games

Georgios D. Varsamis, Ioannis Liliopoulos, Hamed Mohammadbagherpoor +6 more

Quantum games extend classical game theory into the quantum realm by incorporating quantum players and quantum strategies. Two-player quantum games have been widely studied and applied, among other areas, in optimization and decision-making. In this work, we introduce, formalize and analyze a novel class of three-player quantum games, referred to as Proxy Quantum Games (PQGs). In this setting, one player, the Controller, influences or governs the strategic operations executed by a second player, the Proxy, who, in turn, competes directly against a third player, the Opponent. The strategic influence is implemented via controlled unitary operations, whereby the Proxy’s available strategies are conditioned on the Controller’s quantum state. We further generalize the PQG framework by incorporating multiple Controllers acting on a common Proxy, thereby enabling the study of higher-order strategic interactions and multi-agent influence structures. The objective under consideration is the maximization of the Controller’s expected payoff, under various configurations of strategic control and entanglement. Comparative analyses are conducted between the PQG formulation and conventional multiplayer quantum games, highlighting key differences in strategic behavior and payoff distributions.

Parallel Processing Letters•Sep 1, 2025•1 citation
Abstract Entanglement is one of the most vital resources in quantum computing. Particularly, in quantum game theory, entanglement holds significant importance since it defines the interactions and the information exchange between two or more players. In this research work, a new set of entanglement operators is introduced for N‐player quantum games. This study explores how their application affects each player's behavior throughout the game. Furthermore, the entangling capabilities are tested and evaluated for a wide range of entanglement operator angles, by exploiting the Von Neumann entropy metric, both on a simulator and on a real IBM Quantum system. Their hardware efficiency, against state‐of‐the‐art quantum games entanglement operators, is assessed in terms of circuit depths and fidelity, and showcases their great potential as multiplayer quantum games operators. Finally, three methods are compared for computing the players’ payoffs, regarding their scalability and efficiency. This study shows that in all payoff cases, the quantum game results, computed by the quantum hardware, are in accordance with the simulated ones.
In quantum games based on 2-player-N-strategies classical games, each player has a quNit (a normalized vector in an N-dimensional Hilbert space HN) upon which he applies his strategy (a matrix U∈SU(N)). The players draw their payoffs from a state . Here and J (both determined by the game’s referee) are respectively an unentangled 2-quNit (pure) state and a unitary operator such that is partially entangled. The existence of pure strategy Nash equilibrium in the quantum game is intimately related to the degree of entanglement of . Hence, it is practical to design the entangler J= J(β) to be dependent on a single real parameter β that controls the degree of entanglement of , such that its von-Neumann entropy SN(β) is continuous and obtains any value in . Designing J(β) for N=2 is quite standard. Extension to N>2 is not obvious, and here we suggest an algorithm to achieve it. Such construction provides a special quantum gate that should be a useful tool not only in quantum games but, more generally, as a special gate in manipulating quantum information protocols.
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 what
In this paper, we extend the quantum game theory of Prisoner’s Dilemma to the N-player case. The final state of quantum game theory of N-player Prisoner’s Dilemma is derived, which can be used to investigate the payoff of each player. As demonstration, two cases (2-player and 3-player) are studied to illustrate the superiority of quantum strategy in the game theory. Specifically, the non-unique entanglement parameter is found to maximize the total payoff, which oscillates periodically. Finally, the optimal strategic set is proved to depend on the selection of initial states.
We present a general two-player quantum game simulator that can simulate any two-player quantum game described by a 2times2 payoff matrix (two strategy games).The user can determine the payoff matrices for both players, their strategies and the amount of entanglement between their initial strategies. The outputs of the simulator are the expected payoffs of each player as a function of the other player's strategy parameters and the amount of entanglement. The simulator also produces contour plots that divide the strategy spaces of the game in regions in which players can get larger payoffs if they choose to use a quantum strategy against any classical one. We also apply the simulator to two well known quantum games, the Battle of Sexes and the Chicken game.
Quantum game theory is a multidisciplinary field which combines quantum mechanics with game theory by introducing non-classical resources such as entanglement, quantum operations and quantum measurement. By transferring two-player two-strategy (2 × 2) dilemma containing classical games into the quantum realm, dilemmas can be resolved in quantum pure strategies if entanglement is distributed between the players who use quantum operations. Moreover, players receive the highest sum of payoffs available in the game, which are otherwise impossible in classical pure strategies. Encouraged by the observation of rich dynamics of physical systems with many interacting parties and the power of entanglement in quantum versions of 2 × 2 games, it became generally accepted that quantum versions can be easily extended to N -player situations by simply allowing N -partite entangled states. In this article, however, we show that this is not generally true because the reproducibility of classical tasks in the quantum domain imposes limitations on the type of entanglement and quantum operators. We propose a benchmark for the evaluation of quantum and classical versions of games, and derive the necessary and sufficient conditions for a physical realization. We give examples of entangled states that can and cannot be used, and the characteristics of quantum operators used as strategies.
Quantum games extend classical game theory into the quantum realm by incorporating quantum players and quantum strategies. Two-player quantum games have been widely studied and applied, among other areas, in optimization and decision-making. In this work, we introduce, formalize and analyze a novel class of three-player quantum games, referred to as Proxy Quantum Games (PQGs). In this setting, one player, the Controller, influences or governs the strategic operations executed by a second player, the Proxy, who, in turn, competes directly against a third player, the Opponent. The strategic influence is implemented via controlled unitary operations, whereby the Proxy’s available strategies are conditioned on the Controller’s quantum state. We further generalize the PQG framework by incorporating multiple Controllers acting on a common Proxy, thereby enabling the study of higher-order strategic interactions and multi-agent influence structures. The objective under consideration is the maximization of the Controller’s expected payoff, under various configurations of strategic control and entanglement. Comparative analyses are conducted between the PQG formulation and conventional multiplayer quantum games, highlighting key differences in strategic behavior and payoff distributions.