Quantum Game Research Articles

Reducing food loss and waste in a two-echelon food supply chain: A quantum game approach

This paper proposes a quantum game to study the food loss and waste (FLW) reduction in a two-echelon food supply chain (FSC) consisting of single supplier and single retailer. First, a Non-zero-sum game model based on the efforts of supplier and retailer in the process of FLW reduction is developed. Then, the classic strategy space is extended to quantum strategy space. The results show that in both classical environment and the separable quantum game scenario, it is difficult to achieve the Pareto optimal strategy of the two parties adopting the full effort strategy, because the full-effort party will bear the risk of betrayal by the non-effort party. However, in the context of maximally entangled quantum game, the risk causing by the non-effort party are borne by himself rather than the full-effort party. Correspondingly, both parties will adopt the full effort strategy to achieve a win-win situation and improve FLW in FSC significantly. Furthermore, an entanglement contract is proposed to ensure that neither of them has the motivation to deviate from the quantum strategy. Based on these findings, some managerial implications are presented to improve the level of cooperation and effort of the supplier and the retailer on FLW reduction.

Surpassing the classical limit in magic square game with distant quantum dots coupled to optical cavities

The emergence of quantum technologies is heating up the debate on quantum supremacy, usually focusing on the feasibility of looking good on paper algorithms in realistic settings, due to the vulnerability of quantum systems to myriad sources of noise. In this vein, an interesting example of quantum pseudo-telepathy games that quantum mechanical resources can theoretically outperform classical resources is the Magic Square game (MSG), in which two players play against a referee. Due to noise, however, the unit winning probability of the players can drop well below the classical limit. Here, we propose a timely and unprecedented experimental setup for quantum computation with quantum dots inside optical cavities, along with ancillary photons for realizing interactions between distant dots to implement the MSG. Considering various physical imperfections of our setup, we first show that the MSG can be implemented with the current technology, outperforming the classical resources under realistic conditions. Next, we show that our work gives rise to a new version of the game. That is, if the referee has information on the physical realization and strategy of the players, he can bias the game through filtered randomness, and increase his winning probability. We believe our work contributes to not only quantum game theory, but also quantum computing with quantum dots.

Board games for quantum computers

Scalable board games, including Five in a Row (or gomoku) and weiqi (or go), are generalized so that they can be played on or by quantum computers. We adopt three principles for the generalization: the first two are to ensure that the games are compatible with quantum computer and the third is to ensure that the standard classical games are the special cases. We demonstrate how to construct basic quantum moves and use them to set up quantum games. There are three different schemes to play the quantized games: one quantum computer with another quantum computer (QwQ), two classical computer playing with each other on one quantum computer (CQC), and one classical computer with another classical computer(CwC). We illustrate these results with the games of Five in a Row and weiqi.

Simultaneous certification of entangled states and measurements in bounded dimensional semiquantum games

Certification of quantum systems and operations is a central task in quantum information processing. Most current schemes rely on a tomography with fully characterized devices, while this may not be met in real experiments. Device characterizations can be removed with device-independent tests, but it is technically challenging at the moment. In this paper, we investigate the problem of certifying entangled states and measurements via semiquantum games, a type of nonlocal quantum games with well characterized quantum inputs, balancing practicality and device independence. We first design a specific bounded-dimensional semiquantum game, with which any pure entangled state and Bell state measurement operators can be simultaneously certified. Afterward via a duality treatment of state and measurement, we interpret the dual form of this game as a source-independent bounded-dimensional entanglement swapping protocol and show the whole process, including any entangled projector and Bell states, can be certified with this protocol. In particular, our results do not require a complete Bell state measurement, which is beneficial for experiments and practical use.Received 11 February 2020Accepted 12 August 2020DOI:https://doi.org/10.1103/PhysRevResearch.2.033400Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasEntanglement detectionNonlocalityQuantum correlations in quantum informationQuantum entanglementQuantum Information

The chaotic dynamics of a quantum Cournot duopoly game with bounded rationality

This paper analyzes the chaotic dynamics of a quantum Cournot duopoly game with bounded rational players by applying quantum game theory. We investigate the impact of quantum entanglement on the stability of the quantum Nash equilibrium points and chaotic dynamics behaviors of the system. The result shows that the stability region decreases with the quantum entanglement increasing. The adjustment speeds of bounded rational players can lead to chaotic behaviors, and quantum entanglement accelerates the bifurcation and chaos of the system. Numerical simulations demonstrate the chaotic features via stability region, bifurcation, largest Lyapunov exponent, strange attractors, sensitivity to initial conditions and fractal dimensions.

Experimental observation of phase-transition-like behavior in an optical simulation of single-qubit game

Phase-transition-like behavior in quantum games has been studied in recent years as a strategy to deal with abrupt changes in the expected payoff of quantum players. More precisely, such behavior is investigated in the context of two-qubit quantum games, where entanglement is used as a resource for quantum players or quantum game referees. In this work, we present the first experimental realization of a single-qubit game where phase-transition-like behavior raises as an immediate consequence of quantum coherence, since no entanglement is present in our game. The game was realized by encoding the qubit in the polarization degree of freedom of an intense laser beam. We investigate the optimal gain of the quantum player as well as a discussion and obtaining of critical points that was experimentally observed, where we had an excellent agreement between theoretical predictions and the experimental results. Due to the power of the quantum-classical analogy between quantum states and optical modes, our platform is adequate to simulate the quantum game and our experiment can be efficiently reproduced in a single-photon experiment.

Quantum game of Go could baffle even an AI

Quantum game of Go could baffle even an AI

Parrondo paradoxical walk using four-sided quantum coins.

Two losing games can be played in a certain manner to produce a winning outcome-a phenomenon known as Parrondo's paradox. Of particular interest is the emergence of quantum game theory and the attempt to model known Parrondo's games through quantum computation notation. In this article, we investigate whether flipping four-sided quantum coins will result in the emergence of Parrondo's paradox. We discover that by playing two losing games A and B in a sequential order, a winning scenario can be derived. Furthermore, four-sided quantum coin is the first instance where the ratcheting effect from the classical Parrondo's game is necessary. Crucially, our study is designed with quantum protocols as its basis and does not have a direct classical counterpart.

The dynamics of a quantum Bertrand duopoly with differentiated products and heterogeneous expectations

Based on the bounded rationality and naïve expectation, the paper sets up a dynamic model of quantum Bertrand duopoly game with differentiated products by applying quantum game theory. We analyze the effect of quantum entanglement on stability of the quantum equilibrium points and complex dynamic behaviors. The result shows that the quantum entanglement can enhance the stability of the system. The adjustment speeds of bounded rational player can lead to chaotic characteristics of the system, the quantum entanglement can control the system’s chaos. Numerical simulations demonstrate the chaotic features via bifurcation, largest Lyapunov exponent, strange attractors, fractal dimensions and sensitivity to initial conditions.

Machine-learning-based three-qubit gate design for the Toffoli gate and parity check in transmon systems

We use machine learning techniques to design a 50 ns three-qubit flux-tunable controlled-controlled-phase gate with fidelity of >99.99% for nearest-neighbor coupled transmons in circuit quantum electrodynamics architectures. We explain our gate design procedure where we enforce realistic constraints, and analyze the new gate's robustness under decoherence, distortion, and random noise. Our controlled-controlled-phase gate in combination with two single-qubit gates realizes a Toffoli gate which is widely used in quantum circuits, logic synthesis, quantum error correction, and quantum games.

Von Neumann's Minimax Theorem for Continuous Quantum Games

The concept of a classical player, corresponding to a classical random variable, is extended to include quantum random variables in the form of self adjoint operators on infinite dimensional Hilbert space. A quantum version of Von Neumann's Minimax theorem for infinite dimensional (or continuous) games is proved.

Quantum Analysis on Task Allocation and Quality Control for Crowdsourcing With Homogeneous Workers

Crowdsourcing has been emerging as a valid problem-solving model that harnesses a large group of contributors to solve a complicated task. However, existing crowdsourcing platforms or systems could suffer from task allocation and quality control problems. In this article, we first prove that there exist two dilemmas while tackling the above issues by using a game-theoretic approach. To overcome this challenge, we are focusing on exploiting quantum crowdsourcing schemes in which the welfare of requestor or worker can be maximized since quantum players share the extended strategy space, and the introduction of entanglement offers a new method of depicting fine-grained relations between players. Specifically, we propose a quantum game model for quota-oriented crowdsourcing game to address dilemmas in task allocation. The result indicates the dilemma based on classical strategy will disappear with the increment of entanglement degree. While in the quality-oriented crowdsourcing game, we adopt a density matrix approach to calculate the optimal payoffs of both sides, which demonstrates the superiority of our quantum strategy. Moreover, our quantum scheme is generic since it is compatible with the schemes from a classical perspective. Hence, our noteworthy quantum crowdsourcing schemes offer a promising alternative route for tackling dilemmas in crowdsourcing scenarios.

Quantum Key‐Distribution Protocols Based on a Quantum Version of the Monty Hall Game

AbstractThis work illustrates a possible application of quantum game theory to the area of quantum information, in particular to quantum cryptography. The study proposed two quantum key‐distribution (QKD) protocols based on the quantum version of the Monty Hall game devised by Flitney and Abbott. Unlike most QKD protocols, in which the bits from which the key is going to be extracted are encoded in a basis choice (as in BB84), these are encoded in an operation choice. The first proposed protocol uses qutrits to describe the state of the system and the same game operators proposed by Flitney and Abbott. The motivation behind the second proposal is to simplify a possible physical implementation by adapting the formalism of the qutrit protocol to use qubits and simple logical quantum gates. In both protocols, the security relies on the violation of a Bell‐type inequality, for two qutrits and for six qubits in each case. Results show a higher ratio of violation than the E91 protocol.

Bistable probabilities: a unified framework for studying rationality and irrationality in classical and quantum games.

This article presents a unified probabilistic framework that allows both rational and irrational decision-making to be theoretically investigated and simulated in classical and quantum games. Rational choice theory is a basic component of game-theoretic models, which assumes that a decision-maker chooses the best action according to their preferences. In this article, we define irrationality as a deviation from a rational choice. Bistable probabilities are proposed as a principled and straightforward means for modelling (ir)rational decision-making in games. Bistable variants of classical and quantum Prisoner's Dilemma, Stag Hunt and Chicken are analysed in order to assess the effect of (ir)rationality on agent utility and Nash equilibria. It was found that up to three Nash equilibria exist for all three classical bistable games and maximal utility was attained when agents were rational. Up to three Nash equilibria exist for all three quantum bistable games; however, utility was shown to increase according to higher levels of agent irrationality.

Modeling Coopetition as a Quantum Game

Coopetition is defined as the existence of simultaneous competition and cooperation between the same set of players, leading to entanglement of payoffs and actions of the players. This paper provides insights into the game theoretical application of quantum games to model simultaneity and entanglement that occur in coopetition. Modeling as a quantum game also allows for a larger action space and hence new equilibrium that may not have existed earlier. The impact of the level of entanglement on the equilibrium of the game can also be studied. We demonstrate the same through an example of two players who currently compete in the domestic market and are considering cooperating simultaneously in the international market. They need to determine the equilibrium strategy to adopt under coopetition that maximizes their payoffs. We also arrive at how to ensure that the quantum strategy is the equilibrium strategy for both players, namely, how to design the quantum strategy and how to define the unitary operator.

Monogamy relations for the generalized W -class states beyond qubits

In this article, we consider the monogamy relations for the generalized $W$-class states. Here we first present an analytical formula on Tsallis-$q$ entanglement ($\mathrm{T}q\mathrm{EE}$) and Tsallis-$q$ entanglement of assistance ($\mathrm{T}q\mathrm{EEoA}$) of a reduced density matrix for a generalized $W$-class state. By using the analytical formula, we present a monogamy relation in terms of the squared $\mathrm{T}q\mathrm{EE}$ for a generalized $W$-class state. Then we present generalized monogamy relations in terms of $\mathrm{T}q\mathrm{EE}$ for a generalized $W$-class state. We also present a monogamy relation in terms of an arbitrary power of $\mathrm{T}q\mathrm{EE}$ for a generalized $W$-class state. Last, we apply our results to quantum games and present the bound of the nonclassicality of the quantum games when restricting to the generalized $W$-class states.

Perfect Prediction in normal form: Superrational thinking extended to non-symmetric games

This paper introduces a new solution concept for non-cooperative games in normal form with no ties and pure strategies: the Perfectly Transparent Equilibrium. The players are rational in all possible worlds and know each other’s strategies in all possible worlds — which, together, we refer to as Perfect Prediction. The anticipation of a player’s decision by their opponents is counterfactually dependent on the decision, unlike in Nash Equilibria where the decisions are made independently. The equilibrium, when it exists, is unique and is Pareto optimal.This equilibrium is the normal-form counterpart of the Perfect Prediction Equilibrium; the prediction happens “in another room” rather than in the past. The equilibrium can also be seen as a natural extension of Hofstadter’s superrationality to non-symmetric games. Algorithmically, an iterated elimination of non-individually-rational strategy profiles is performed until at most one remains. An equilibrium is a strategy profile that is immune against knowledge of strategies in all possible worlds and rationality in all possible worlds, a stronger concept than common knowledge of rationality but also stronger than common counterfactual belief of rationality.We formalize and contrast the Non-Nashian Decision Theory paradigm, common to this and several other papers, with Causal Decision Theory and Evidential Decision Theory. We define the Perfectly Transparent Equilibrium algorithmically and prove (when it exists) that it is unique, that it is Pareto-optimal, and that it coincides with Hofstadter’s Superrationality on symmetric games. We relate the finding to concepts found in the literature such as Individual Rationality, Rationalizability, Minimax-Rationalizability, Second-Order Nash Equilibria, the Program Equilibrium, the Perfect Prediction Equilibrium, Shiffrin’s Joint-Selfish-Rational Equilibrium, the Stalnaker–Bonanno Equilibrium, the Perfect Cooperation Equilibrium, the Translucent Equilibrium, the Correlated Equilibrium, and Quantum Games. Finally, we specifically discuss inclusion relationships on the special case of symmetric games between Individual Rationality, Minimax-Rationalizability, Superrationality, and the Perfectly Transparent Equilibrium, and contrast them with asymmetric games.

Implementation of sequential game on quantum circuits

It is known that Eisert–Wilkens–Lewenstein scheme and Marinatto–Weber scheme are widely used to quantize simultaneous games. Further, distributed quantum game can be implemented in quantum networks using the schemes based on client–server model and peer-to-peer model. While most of the research works focus on simultaneous games, in this work, we propose the quantum circuit implementation of sequential game, namely cop and robber game using peer-to-peer model. Thus we have made a modest attempt to show the possibility of implementing sequential games in quantum circuits. Further, quantum version of cop and robber game is analyzed. Implications of the results are discussed from the game-theoretic point of view and quantum circuit point of view.

Preferences in quantum games

A quantum game can be viewed as a state preparation in which the final output state results from the competing preferences of the players over the set of possible output states that can be produced. It is therefore possible to view state preparation in general as being the output of some appropriately chosen (notional) quantum game. This reverse engineering approach in which we seek to construct a suitable notional game that produces some desired output state as its equilibrium state may lead to different methodologies and insights. With this goal in mind we examine the notion of preference in quantum games since if we are interested in the production of a particular equilibrium output state, it is the competing preferences of the players that determine this equilibrium state. We show that preferences on output states can be viewed in certain cases as being induced by measurement with an appropriate set of numerical weightings, or payoffs, attached to the results of that measurement. In particular we show that a distance-based preference measure on the output states is equivalent to a having a strictly-competitive set of payoffs on the results of some measurement.

Quantum pseudo-telepathy in spin systems: the magic square game under magnetic fields and the Dzyaloshinskii–Moriya interaction

The Dzyaloshinskii–Moriya (DM) interaction has been proved to excite entanglement of spin systems, enhancing the ability to realizing various quantum tasks such as teleportation. In this work we consider DM interaction—to the best of our knowledge for the first time—in quantum game theory. We study the winning probability of the magic square game played with the thermally entangled state of spin models under external magnetic fields and DM interaction. We analytically show that although DM interaction excites the entanglement of the system as expected, it surprisingly reduces the winning probability of the game, acting like temperature or magnetic fields, and also show that the effects of DM interaction and an inhomogeneous magnetic field on the winning probability are identical. In addition, we show that the XXZ model is considerably more robust than the XX model against destructive effects. Our results open up new insights for quantum information processing with spin systems.