An Exploration of Quantum Game Theory and its Applications Khaled C. Allen Department of Mathematics Defense Date: October 29, 2020 Undergraduate Honors Thesis University of Colorado Boulder Thesis Advisor: Dr. Divya Vernerey, Department of Mathematics Defense Committee: Dr. Divya Vernerey, Department of Mathematics Dr. Nathaniel Thiem, Department of Mathematics Dr. Graeme Baird Smith, Department of Physics 1 Abstract Quantum computation has provided algorithms that outperform classical algorithms in a variety of fields, and recently, game theory has been added to that list [7]. It has been shown that a quantum strategy dominates classical strategies in examples such as a simple coin-flip game [7] and the prisoner's dilemma [5]. Eisert, Wilkens, and Lewenstein have provided a model to quantize general 2-player games[5]. Dahl and Landsburg explore the application of quantum randomization devices to player strategies in classical two-person games [4]. Many problems in quantum computer science, such as Grover's search and Simon's prob- lem, can be modeled as games [7], and the quantum algorithms that optimally solve them seen as player moves in a game state. Additionally, error correction [7] and cryptographic problems lend themselves well to modeling as games [10]. In many of these cases, modeling the problem as a game enables actors to overcome inefficiencies caused by lack of communication, as in the prisoner's dilemma [5] and in a simple cooperative coordination game proposed for the GHZ Paradox [6]. Since quantum algorithm solutions to problems can be translated into games, this begs the question of whether modeling more problems as games can lead to the construction of an algorithm to solve them. Additionally, the potential for quantum systems to provide new means to coordinate action merits further exploration. This thesis is both an introduction to, and a review of some of the existing literature on, quantum game theory. As such, we provide additional details of the computations done in the papers under review in hopes of elucidating the overall process of constructing and analyzing quantum games. Also, we develop and present a new game that extends [7], and applies analytic techniques from [5]. A Q# programming language implementation of this new game is also provided. 1 Contents 1 Abstract 1 2 Background4 2.1 Overview......................................4 3 General Theory of Quantum Games5 3.1 Formal Description of a Quantum Game.....................5 3.1.1 Quantum Moves..............................6 3.1.2 Payout Functions..............................7 3.2 Game Theory....................................7 3.2.1 Nash Equilibria...............................8 3.2.2 Pure and Mixed Strategies........................8 4 Examples of Quantum Games9 4.1 PQ Penny Flip...................................9 4.1.1 Setup of the Quantum Coin Flip Game.................9 4.1.2 The Classical Deterministic Strategy................... 10 4.1.3 The Classical Probabilistic Strategy................... 10 4.1.4 Quantum Strategy, Picard's Classical Moves............... 10 4.1.5 Optional: Calculating Payout....................... 11 4.1.6 Quantum Strategy, Q's Quantum Moves................. 11 4.1.7 Quantum Strategy Analysis........................ 12 4.2 Prisoner's Dilemma................................. 12 4.2.1 The Quantum Prisoner's Dilemma.................... 13 4.2.2 Available Moves.............................. 13 4.2.3 Measurements............................... 13 4.2.4 Taking a Turn............................... 14 4.2.5 Calculating Payout............................. 14 4.2.6 The General Case............................. 15 4.2.7 Calculating the Trace........................... 15 4.2.8 General Payout............................... 16 4.2.9 A New Nash Equilibrium......................... 17 4.3 Zero-Sum Simultaneous PQ Penny Flip..................... 17 4.3.1 Classical Strategic Space.......................... 19 4.3.2 Restricted Quantum Strategic Space................... 19 4.3.3 The Full Bloch Sphere Strategic Space.................. 19 4.3.4 Simultaneous PQ Penny Flip with PD Starting State and Move Set. 22 5 A Quantum Game Theoretic Approach to a Communication Problem 25 5.1 Classical Strategies Fail.............................. 25 5.2 Quantum Strategy................................. 26 6 Conclusion and Extensions 27 7 Acknowledgements 28 2 A Fundamentals of Quantum Computing 29 A.1 Qubits........................................ 29 A.2 Diract Notation and Vector Notation....................... 29 A.3 The Math of Measurement............................. 31 A.4 Operations on Qubits............................... 32 A.5 Tensor Products and Qubit Registers....................... 33 B Q# Implementation of Simultaneous PQ Penny Flip 34 C Mathematica Notebooks for Calculations and Payout Graphs 35 D Bibliography 44 3 2 Background Since its formalization by von Neumann and Morgenstern [8], game theory has proven a valuable tool in modeling phenomena in economics, computer science, physics, biology, and others [7, 10]. As these fields have progressed, they have had to contend with the quantum nature of phenomena on small scales. Thus, game theory stands to be generalized to deal with the nature of information exchange and feedback on quantum scales [7]. Quantum computation has provided algorithms that outperform classical algorithms in a variety of fields, and recently, game theory has been added to that list [7]. For example, it has been shown that a quantum strategy dominates classical strategies in a simple coin-flip game [7] and the prisoner's dilemma [5]. Eisert, Wilkens, and Lewenstein have provided a model to quantize general 2-player games [5]. Dahl and Landsburg have explored the application of quantum randomization devices to player strategies in classical two-person games [4]. Many problems in quantum computer science can be modeled as games, such as Grover's search and Simon's problem [7], and the quantum algorithms that optimally solve them can be seen as player moves in a game state. Additionally, error correction [7] and cryptographic and communication problems (the GHZ1 puzzle at the end of this paper being an example of the latter) lend themselves well to modeling as games [10, 12]. Since quantum algorithm solutions to problems can be translated to games, this begs the question of whether modeling problems as games can lead to the construction of an algorithm to solve them. 2.1 Overview This thesis is primarily concerned with three goals: 1. compiling and demonstrating a general theory for the formulation and analysis of quantum games, 2. examining several illustrative examples of quantum game theory, and 3. developing a quantum game in order to analyze it using established methods and model it using a quantum computing language, Q#. Additionally, we briefly explore how modeling a communication problem as a quantum game provides a fruitful method of solving it. The general theory for quantum games comes from various sources in the literature, particularly [5,7,2]. Each has provided some exploration of fundamental quantum game theoretic principles, and we construct a cohesive, universal theory that applies to all the games under analysis. Then, we explore specific quantum games: the PQ Penny Flip developed by [7]; the Quantum Prisoner's Dilemma developed by [5]; and an extended version of the PQ Penny Flip we have modified to explore implications for a simple zero-sum quantum game. In each case, we provide details of the calculations that have been left out of the published works to elucidate the underlying process of calculating game theoric equilibria, as well as to facilitate 1Named after Daniel Greenberger, Michael Horne, and Anton Zeilinger, who discovered the related paradox in the late 1980s [6]. 4 modeling games with Q#. We also present a Q# implementation of the Simultaneous PQ Penny Flip that can be run and played on a classical computer. The remainder of the paper examines how situations that are not generally thought of as games can be modeled as such, specifically looking at the GHZ Paradox coordination challenge from the perspective of quantum game theory. The goal is to determine if viewing these problems through the lens of a game may have suggested the approaches used to solve them, or suggests alternate approaches. Appendices contain some background information on the quantum computing theory necessary to understand the topics presented. Additionally, the code for the Q# implemen- tation of the Simultaneous PQ Penny Flip and the Mathematica code used for generating the graphs in the paper are provided in the appendices. 3 General Theory of Quantum Games A classical game consists of a set of players, a set of actions each player may take, and a preference relation that ranks the game's outcomes for each player. Definition 3.1. A game consists of the set, N, of the players, a set Ai of actions each player may take for i 2 N, and a preference relation, ≥, for each player on the set A = ×i2N Ai[9]. The preference relation can be thought of as a payoff function by allowing player i to rank the various outcomes of the game. Eisert provides a straightforward definition of a quantum game: \Any quantum system which can be manipulated by two parties or more and where the utility of the moves can be reasonably quantified, may be conceived of as a quantum game" [5]. In conceptual terms, a quantum game is a quantum system (see AppendixA) that the players can manipulate, and which has certain payouts to each player associated