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Investigation into the effects of noise on quantum strategies

Wee Ling Ang

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

Research in the field of quantum computation, in particular, quantum game theory is in its foundational stage. Primarily, quantum game theory adds a new dimension to the well - developed classical game theory by making possible the superposition of strategies. By projecting the classical Prisoner's Dilemma game into the quantum domain, experts in this regime proved that, for an ideally closed quantum system, superposed quantum strategies offer more promising alternatives and solutions than discrete classical ones for this peculiar game. To present a more realistic picture of a quantum game system, this project seeks to examine the efficiency of quantum strategies in a less than perfect environment where decoherence of pure quantum states can occur. The findings advocate that quantum strategies remain superior to classical ones despite the deterioration of solutions due to decoherence.

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Research in the field of quantum computation, in particular, quantum game theory is in its foundational stage. Primarily, quantum game theory adds a new dimension to the well - developed classical game theory by making possible the superposition of strategies. By projecting the classical Prisoner's Dilemma game into the quantum domain, experts in this regime proved that, for an ideally closed quantum system, superposed quantum strategies offer more promising alternatives and solutions than discrete classical ones for this peculiar game. To present a more realistic picture of a quantum game system, this project seeks to examine the efficiency of quantum strategies in a less than perfect environment where decoherence of pure quantum states can occur. The findings advocate that quantum strategies remain superior to classical ones despite the deterioration of solutions due to decoherence.
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
Theory of games is an important discipline of applied mathematics, which has found numerous applications in psychology, ecology, and social science. While a well-established framework has been established for classical game theory, extending, and generalizing it towards the quantum domain still has some open questions. In this project, we focus on the prisoner’s dilemma-a standard example of a game analyzed in game theory-in a quantum world. Here, we start with the quantum version of prisoner’s dilemma, explain the quantum circuit implementation for describing the behavior of quantum prisoners, and investigate how quantum entanglement can change their strategies. By using the IBM quantum computer, we experimentally study the quantum prisoner dilemma, and we conclude that the best quantum strategies will break the prisoner’s dilemma.
This work concentrates on simultaneous move non-cooperating quantum games. Part of it is evidently not new, but it is included for the sake self consistence, as it is devoted to introduction of the mathematical and physical grounds of the pertinent topics, and the way in which a simple classical game is modified to become a quantum game (a procedure referred to as a quantization of a classical game). The connection between game theory and information science is briefly stressed, and the role of quantum entanglement (that plays a central role in the theory of quantum games), is exposed. Armed with these tools, we investigate some basic concepts like the existence (or absence) of a pure strategy and mixed strategy Nash equilibrium and its relation with the degree of entanglement. The main results of this work are as follows: 1) Construction of a numerical algorithm based on the method of best response functions, designed to search for pure strategy Nash equilibrium in quantum games. The formalism is based on the discretization of a continuous variable into a mesh of points, and can be applied to quantum games that are built upon two-players two-strategies classical games, based on the method of best response functions. 2) Application of this algorithm to study the question of how the existence of pure strategy Nash equilibrium is related to the degree of entanglement (specified by a continuous parameter γ ). It is shown that when the classical game GC has a pure strategy Nash equilibrium that is not Pareto efficient, then the quantum game GQ with maximal entanglement (γ = π/2) has no pure strategy Nash equilibrium. By studying a non-symmetric prisoner dilemma game, it is found that there is a critical value 0γc such that for γγc there is a pure strategy Nash equilibrium and for γ≥γc there is no pure strategy Nash equilibrium. The behavior of the two payoffs as function of γ starts at that of the classical ones at (D, D) and approaches the cooperative classical ones at (C, C) (C = confess, D = don’t confess). 3) We then study Bayesian quantum games and show that under certain conditions, there is a pure strategy Nash equilibrium in such games even when entanglement is maximal. 4) We define the basic ingredients of a quantum game based on a two-player three strategies classical game. This requires the introduction of trits (instead of bits) and quantum trits (instead of quantum bits). It is proved that in this quantum game, there is no classical commensurability in the sense that the classical strategies are not obtained as a special case of the quantum strategies.
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 have demonstrated to propose a new point of view for the solution of the classical problems and dilemmas in game theory. It has been shown in many literatures that quantum games are more efficient than classical games and provide a greater and more efficient solution. As interest in multi-player quantum games grow, methods are proposed to analyze and obtain efficient solutions for these problems. In quantum games with cooperation, analysis so far have taken into consideration all possible scenarios for the multiple players playing different strategies and in the calculation of payoff, setting the payoff of scenarios not possible to be zero. Chapter 5 presents the analysis of multiple players (three players and four players) using this method of analysis. In Chapter 6, a modification of the analysis was done. In Chapter 5, it should be noted that the probabilities of participating scenarios do not sum to unity which is technically not ideal. Hence Chapter 6 addresses this by performing re-normalizing the probabilities of the participating scenarios so that the probabilities sum to unity. The analysis was done for three and four player games and a comparison was made between the un-normalized method and the re-normalized method. It was found that re-normalization provides a more accurate representation of the classical game and hence can be a better solution for the quantum game.