Quantum Game Research Articles

Nash Equilibria and Undecidability in Generic Physical Interactions—A Free Energy Perspective

We start from the fundamental premise that any physical interaction can be interpreted as a game. To demonstrate this, we draw upon the free energy principle and the theory of quantum reference frames. In this way, we place the game-theoretic Nash Equilibrium in a new light in so far as the incompleteness and undecidability of the concept, as well as the nature of strategies in general, can be seen as the consequences of certain no-go theorems. We show that games of the generic imitation type follow a circularity of idealization that includes the good regulator theorem, generalized synchrony, and undecidability of the Turing test. We discuss Bayesian games in the light of Bell non-locality and establish the basics of quantum games, which we relate to local operations and classical communication protocols. In this light, we also review the rationality of gaming strategies from the players’ point of view.

Citizen science games on the timeline of quantum games

This article provides an overview of existing quantum physics-related games, referred to as quantum games, that serve citizen science research in quantum physics. Additionally, we explore the connection between citizen science and quantum computer games, games played on quantum computers. The information presented is derived from academic references and supplemented by diverse sources, including social media publications, conference presentations and blog posts from research groups and developers associated with the presented games. We observe that the current landscape of quantum games is shaped by three distinct driving forces: the serious application of games, the evolution of quantum computers and open game development events such as Quantum Game Jams. Notably, citizen science plays an influential role in all three aspects. The article points to existing design guides for citizen science quantum games and views future prospects of citizen science projects and quantum games through collaborative endeavours, human–machine collaboration and open access quantum computers.

A Chronicle of Quantum Technologies in Game and Software Development.

The ongoing second quantum revolution, marked by advancements harnessing quantum phenomena, has permeated various fields, including communications, computation, and networking, collectively known as quantum technologies (QT). Quantum computing, a focal point within QT, has led to the emergence of quantum games, a novel research and development area exploring creative applications of quantum computing in game development. While more than 300 quantum games have been developed by enthusiasts and commercial parties, a comprehensive research program or state-of-the-art review is notably absent. This paper addresses this gap by conducting a thorough literature review, presenting current advancements and examples in quantum games and interactive designs, and exploring future prospects for quantum technologies in game development practices.

Open access publishing and quantum game theory under modified EWL scheme

Game theory deals with the strategies of rational players to obtain the best possible outcomes for a player in a game. Quantum game theory is an extension of classical game theory, where players can adopt quantum strategies to maximize their payoff advantage. In quantum game theory, we have two prominent quantization schemes, namely the Eisert Wilkens-Lewenstein (EWL) scheme and the Marinatto Weber (MW) scheme. Recently, modified EWL scheme is being used to study the game dynamics. In the era of digital information revolution, scientists have unrestricted access to most of the scientific information. Different scientific disciplines prefer different paths of publication either open access or traditional. Taking this as a framework, we model an open-access game played by scientists with open access and traditional publishing as their strategies. In this work, we study these publication patterns using the quantum game theory approach under the modified EWL quantization scheme. Specifically, we investigate the role of entangling operators and players’ strategies in maximizing the payoff (reputation) for the scientists playing in the open-access game.

Complex dynamics and chaos control of a quantum Stackelberg game with heterogeneous players

In this research article, we explore the complex dynamics of a quantum Stackelberg game and its chaos control involving players with heterogeneous expectations. We utilize the Iqbal & Toor scheme to quantize the classical Stackelberg game and observe that the first mover advantage is diminished in the quantum version of the game. By examining the game from an evolutionary perspective, where the leader firm employs bounded rational expectations and the follower firm adopts adaptive expectations, we discover that the quantum Stackelberg game exhibits complex behaviors, such as cyclic and chaotic states. The local stability of the quantum game depends on the initial state of the game. Additionally, the occurrence of chaos is delayed in the quantum game. To further investigate, we implement the time-delayed feedback method to control the chaos in the quantum game and demonstrate that the techniques used for controlling the classical game are still effective in the quantum game.

Exploiting finite geometries for better quantum advantages in Mermin-like games

Quantum games embody non-intuitive consequences of quantum phenomena, such as entanglement and contextuality. The Mermin-Peres game is a simple example, demonstrating how two players can utilise shared quantum information to win a no—communication game with certainty, where classical players cannot. In this paper we look at the geometric structure behind such classical strategies, and borrow ideas from the geometry of symplectic polar spaces to maximise this quantum advantage. We introduce a new game called the Eloily game with a quantum–classical success gap of 0.26― , larger than that of the Mermin-Peres and doily games. We simulate this game in the IBM Quantum Experience and obtain a success rate of 1, beating the classical bound of 0.73― demonstrating the efficiency of the quantum strategy.

Reducing betrayal behavior in green building construction: A quantum game approach

The issue of energy consumption in buildings is becoming increasingly pertinent. Society highly values energy conservation and environmental protection, viewing the adoption of green buildings as an inevitable trajectory in the evolution of the construction sector. However, a challenge emerges during the progression of green building construction concerning betrayal within the supply chain. This paper employs a quantum game approach to tackle the issue of green building betrayal within a two-tier green building supply chain comprising developers and contractors. Initially, a non-zero-sum classical game model is formulated based on the green-building construction process between developers and contractors Subsequently, the classical strategy space is extended to the quantum strategy space. The findings indicate that in classical games, and without considering quantum entanglement, even if both parties adopt comprehensive strategies, the party exerting complete effort still faces the risk of betrayal. Conversely, in scenarios of maximum quantum entanglement, the loss incurred by the party not exerting complete effort does not affect the party making a complete effort. Consequently, when accounting for quantum entanglement, both parties tend to opt for complete effort. Lastly, an entanglement treaty is proposed to constrain the strategic choices of both parties and ensure that neither has an incentive to betray. The findings of this study provide good suggestions for green building project managers to improve the effort degree of the developer and general construction contractor in green buildings to promote the development of green buildings.

Quantum No-signalling Correlations and Non-local Games

We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.

Quantum games under Markovian and non-Markovian quantum noise regime and the nature of phase transitions

Quantum games under Markovian and non-Markovian quantum noise regime and the nature of phase transitions

Non-Hermitian CHSH* Game with a Single Trapped-Ion Qubit

Abstract The Clauser–Horne–Shimony–Holt (CHSH) game provides a captivating illustration of the advantages of quantum strategies over classical ones. In a recent study, a variant of the CHSH game leveraging a single qubit system, referred to as the CHSH* game, has been identified. We demonstrate that this mapping relationship between these two games remains effective even for a non-unitary gate. Here we delve into the breach of Tsirelson’s bound in a non-Hermitian system, predicting changes in the upper and lower bounds of the player’s winning probability when employing quantum strategies in a single dissipative qubit system. We experimentally explore the impact of the CHSH* game on the player’s winning probability in a single trapped-ion dissipative system, demonstrating a violation of Tsirelson’s bound under the influence of parity-time ( P T ) symmetry. These results contribute to a deeper understanding of the influence of non-Hermitian systems on quantum games and the behavior of quantum systems under P T symmetry, which is crucial for designing more robust and efficient quantum protocols.

A Study of Quantum Game for Low-Carbon Transportation with Government Subsidies and Penalties

Traditional classical game theory struggles to effectively address the inefficiencies in subsidizing and penalizing the R&D and production of low-carbon transportation vehicles. To avoid the shortcomings of classic game theory, this research integrates quantum game theory with Nash games to explore the characteristics of automakers’ behavior for low-carbon transportation with government subsidies and penalties. We first constructed a low-carbon transportation game model between the government and automakers. Then, the optimal profit strategies for both parties in a quantum entangled state were analyzed. Finally, the impact of quantum superposition states and the initial entangled state on the profit strategies of both parties was simulated and analyzed using Monte Carlo simulations. We find that under the joint effects of government subsidies and penalties, quantum game states and the initial quantum entangled state have a crucial influence on the game’s outcomes. They can encourage the realization of Nash equilibrium and perfect coordination in the quantum game, significantly increasing the profits for both parties. This in turn effectively stimulates automakers to research and produce low-carbon transportation solutions, promoting the rapid development of low-carbon transportation technology. In theory, this research can enrich the Quantum game for improvements in the Nash equilibrium solution for the government to subsidize and penalize the low-carbon transportation problem. Meanwhile, in practice, it can provide guidance and reference in optimal strategy selection conditions for government policymakers and automakers for low-carbon transportation.

Https://armgpublishing.com/journals/sec/volume-8-issue-1/article-5/

In the 21st century, various socio-economic crises have revealed that traditional economic science and (neo)classical thinking are unable to explain all the complexity of current economic problems, therefore the application of more complex and non-trivial economic concepts is gaining relevance. In addition to behavioral and evolutionary economic thinking, models of quantum economics have been developed in recent years, which allow solving economic problems, using mainly quantum thinking and the principles of quantum physics, in particular particle-wave dualism, the principle of uncertainty, the absence of a subject-object distinction, superposition and confusion. The article addresses 3 research questions (RQ). According to RQ1, the paper finds that quantum economics research is dominated by the following topics: quantum economics, quantum finance, quantum decision making, and quantum game theory. According to these four thematic descriptors, the article carries out a systematic modern review of scientific works in the period from 1978 to 2022 (if only 50 works were published in 1978-1999, then in 2000-2022 already 3430), with an emphasis on the most cited (Google Scholar) English-language journal articles. The analysis showed that articles on this topic are published mainly in journals of a non-economic profile, that the peaks of publication activity occur at times of powerful socio-economic upheavals (for example, the dot.com bubble, the substandard/financial crisis, the European debt crisis, the coronavirus pandemic, etc.). In accordance with RQ2, the article defines the key characteristics of the quantum economic model: in contrast to neoclassical economics (based on mechanistic classical physics, is rational and deterministic, with the help of the invisible hand of the market it leads to a stable equilibrium), the quantum approach, on the contrary, considers the economy as more complex, empirically oriented, uncertain, probabilistic, superpositional, as an archetypal example of a quantum social system that has its own versions of duality, measurement, and entanglement. According to RQ 3, the article determines to what extent quantum economics can update (neo)classical economics (integration of new ontological premises into economic thinking, more experimental and practical approach, connection between the concept of entanglement and sustainable development, management of financial risks based on the concept of quantum probability, rethinking the concept of randomness by quantum probability, introduction of quantum money, equilibrium using quantum games, etc.).

Two new non-equivalent three-qubit CHSH games

In this paper, we generalize to three players the well-known CHSH quantum game. To do so, we consider all possible 3-variables Boolean functions and search among them which ones correspond to a game scenario with a quantum advantage (for a given entangled state). In particular, we provide two new three-player quantum games where, in one case, the best quantum strategy is obtained when the players share a $GHZ$ state, while in the other one, the players have a better advantage when they use a $W$ state as their quantum resource. To illustrate our findings we implement our game scenarios on an online quantum computer and prove experimentally the advantage of the corresponding quantum resource for each game.

Repeated quantum game as a stochastic game: Effects of the shadow of the future and entanglement

We present a systematic investigation of the quantum games, constructed using a novel repeated game protocol, when played repeatedly ad infinitum. We focus on establishing that such repeated games – by virtue of inherent quantum-mechanical randomness – can be mapped to the paradigm of stochastic games. Subsequently, using the setup of two-player–two-action games, we explore the pure reactive strategies belonging to the set of reactive strategies, whose support in the quantum games is no longer countably finite but rather non-denumerably infinite. We find that how two pure strategies fare against each other is crucially dependent on the discount factor (the probability of occurrence of every subsequent round) and how much entangled the quantum states of the players are. We contrast the results obtained with the corresponding results in the classical setup and find fundamental differences between them: e.g, when the underlying game is the prisoner’s dilemma, in the quantum game setup, always-defect strategy can be beaten by the tit-for-tat strategy for high enough discount factor.

Fractional-order quantum game chaotic map and its synchronization with application

In this paper, according to the knowledge of discrete fractional calculus and quantum game chaotic map, we first get the fractional quantum game chaotic map, which describes the dynamics of a quantum duopoly game. Then, the phase space, maximum Lyapunov exponent, and bifurcation diagram of the proposed map are obtained for the purpose of observing dynamic behaviors compared with the original quantum game chaotic map. After that, the synchronous controller is designed and the chaos synchronization of the fractional quantum game chaotic map is observed. Finally, we develop the fractional quantum game chaotic map into the chaotic sequence generator and image encryption scheme.

QESRL: Exploring Selfish Reinforcement Learning for Repeated Quantum Games

A novel unified learning algorithm that seamlessly applies to both classical and quantum non-zero sum games is presented. Building upon the exploring selfish reinforcement learning (ESRL) framework previously proposed in the context of classical games, we extend this approach to handle quantum games with imperfect information. A comparison is made between performance and fairness among agents learning using plain QRL vs. QESRL. The latter enables agents to explore and learn periodic policy strategies in quantum games, leveraging the quantization of games to uncover fairer results. By addressing the challenges posed by the expanded strategy space in quantum games, we test the algorithm’s scalability by increasing the number of agents. Empirical evidence is provided to showcase its performance and to compare classical and quantum game scenarios. The proposed learning algorithm represents a significant step towards understanding the convergence and optimality of strategies in non-zero sum games across classical and quantum settings, bringing us closer to harnessing the potential of independent reinforcement learning and quantum computing in game theory applications.

A Quantum Double-or-Nothing Game: An Application of the Kelly Criterion to Spins.

A quantum game is constructed from a sequence of independent and identically polarised spin-1/2 particles. Information about their possible polarisation is provided to a bettor, who can wager in successive double-or-nothing games on measurement outcomes. The choice at each stage is how much to bet and in which direction to measure the individual particles. The portfolio's growth rate rises as the measurements are progressively adjusted in response to the accumulated information. Wealth is amassed through astute betting. The optimal classical strategy is called the Kelly criterion and plays a fundamental role in portfolio theory and consequently quantitative finance. The optimal quantum strategy is determined numerically and shown to differ from the classical strategy. This paper contributes to the development of quantum finance, as aspects of portfolio optimisation are extended to the quantum realm. Intriguing trade-offs between information gain and portfolio growth are described.

Quantum game— “PQ” problem

This paper reviews the coin flipping problem of “PQ” in a classical two-player game, and shows that when one player adopts quantum strategy, he can beat his classical opponent and gain higher returns. By using quantitative means, the whole “PQ” problem is put under a more general and fair condition, and it is generalized and studied again from many aspects and angles. Finally, we obtain some more essential and important conclusions, and explain the conclusions and its practical significance through some practical examples. At the same time, this paper gives the definition of imperfect fair game and the definition of perfect fair game, revises the quantum coin flipping game to make the game fair, and studies a multi-round version of quantum coin flipping. Some basic conclusions of fair quantum game are obtained, and the meaning of perfectly fair game is explained in practice.

Quantum Game Theory: White paper engineering. The quantum-blockchain platform.

Quantum Game Theory: White paper engineering. The quantum-blockchain platform.

Quantum 2-Player Games and Realizations with Circuits.

Game theory problems are widely applied in many research areas such as computer science and finance, with the key issue being how to quickly make decisions. Here, we present a novel quantum algorithm for game theory problems based on a continuous quantum walk. Our algorithm exhibits quantum advantage compared to classical game algorithms. Furthermore, we exploit the analogy between the wave function of the Schrödinger equation and the voltage in Kirchhoff's law to effectively translate the design of quantum game trees into classical circuit networks. We have theoretically simulated the quantum game trees and experimentally validated the quantum functionality speedup on classical circuit networks. Due to the robust scalability and stability inherent in classical circuit networks, quantum game trees implemented within this framework hold promise for addressing more intricate application scenarios.