Azhar Iqbal

Dr Azhar Iqbal

Research Fellow (A) (with PhD)

School of Computer and Mathematical Sciences

Faculty of Sciences, Engineering and Technology

Eligible to supervise Masters and PhD - email supervisor to discuss availability.

My research spans four domains: quantum game theory, game theory applications, geometric algebra applications, and mathematical modelling.

Quantum Game Theory: My primary research focus lies in the emergent field of quantum game theory, a novel interdisciplinary pursuit that extends the principles of classical game theory into the enigmatic world of quantum mechanics. This pioneering discipline is currently encompassed within the broader quantum information and computation research, where the implications of quantum mechanics are probed and applied to game theory, thereby opening new vistas in its analytical prowess.

Quantum game theory investigates the dynamics of strategic decision-making among rational agents, who possess quantum resources like quantum superposition and entanglement. The objective is to blend the principles of quantum mechanics into conventional game theory scenarios, shedding light on the potential of quantum technology to fundamentally transform our understanding of strategic interactions and decision-making processes.

In this novel realm, a quantum game can be described as strategic interaction among rational agents or 'players', who manipulate a quantum system. Several real-world implementations of quantum games employ local unitary transformations and quantum measurements to encapsulate the strategic manoeuvres and decision-making processes among players.

The payoffs or rewards for each player in a quantum game are shaped by the strategic choices of all participating players and are ultimately derived from the outcomes of quantum measurements. Consequently, the quantum state of the system is impacted by the collective decisions of the players, triggering a complex web of probabilities that dictate each player's payoff. By dissecting these probabilities and the strategic decisions leading to them, quantum game theory can illuminate the intricacies of strategic decision-making in a quantum context. This, in turn, could enable unprecedented applications of quantum technology across various disciplines.

The genesis of quantum games can be traced back to 1999 when David Meyer presented ground-breaking research [,,] illustrating that the quantum algorithm for an oracle problem could be interpreted as a quantum strategy for a player in a two-player zero-sum game. Here, the other player adheres to classical strategy. Meyer's seminal contribution laid the groundwork for quantum game theory, forging a crucial link between quantum computing and game theory that has facilitated continued exploration of quantum technology within the landscape of strategic decision-making.

Soon after Meyer's pioneering work, Eisert, Wilkens, and Lewenstein [,] extended his findings by formulating a quantized interpretation of the famous Prisoners' Dilemma game. This quintessential game theory scenario involves two players who face the strategic dilemma of either cooperating or defecting, with their rewards contingent on the other player's decision. Through their investigation, Eisert, Wilkens, and Lewenstein substantiated the capacity of quantum technology to augment strategic decision-making, underscoring the compelling potential of quantum game theory within the broader domain of quantum information and computation research.

In essence, the capacity to construe quantum algorithms as strategic contests between quantum and classical players has led to the incorporation of game theory into the set of mathematical tools leveraged in the quest to expand the roster of viable quantum algorithms. By assimilating insights from game theory into the design of new quantum algorithms, researchers can acquire a more profound understanding of the intricate relationship between strategic decision-making and quantum mechanics. This, in turn, is instrumental in propelling the ongoing progression of quantum information and computation research.

My scholarly contributions in this domain encompass:

  • The concept of Evolutionarily Stable Strategy (ESS), first introduced by mathematical biologists in the 1970s [,], provides a model for an evolving population under evolutionary pressures. Widely recognized as the cornerstone of evolutionary game theory's stability solution, an ESS mathematically refines the set of symmetric Nash equilibria. My work in this area [,,10 ,11 ,12 ,13 ] delves into the destiny of an ESS when interactions among players within a population, engaged in pairwise symmetric games under evolutionary pressures, adopt quantum mechanics. My research has demonstrated that when pairwise games take on a quantum mechanical nature, quantum entanglement can facilitate the emergence of new Nash equilibria and concurrently influence the future of Nash equilibrium refinements. The manifestation of entanglement can significantly alter the evolution of strategies within a population, indicating potential quantum mechanical foundations for the emergence of self-organization and complexity in molecular-level interactions. A comprehensive review of this topic has been published as a chapter in the book “Quantum Aspects of Life” by the Imperial College Press [14 ];
  • Investigation into the fate of well-established game-theoretic solution concepts within a quantum game has been a focus of my research. This includes exploration of concepts such as "Social Optimality" [15 ], "Value of Coalition" [16 ], "Backwards-Induction Outcome" [17 ], and "Sub-Game Perfect Outcome" [18 ] within the quantum domain;
  • In-depth study of the Einstein-Podolsky-Rosen (EPR) framework for executing quantum games [19 ,20 ], which facilitates the identification of the genuinely quantum components of a quantum game;
  • Development of innovative strategies for creating quantum games, including schemes for quantizing games derived from the concept of non-factorizable joint probabilities [21 ] and from a system of Bell's inequalities [22 ];
  • My recent scholarly pursuits encompass: The introduction of a novel quantization scheme [40 ] wherein each player's quantum strategy is implemented through directional choices across three dimensions. This methodology provides an innovative, geometric, and intuitive means of representing quantum strategies. The formulation of an inventive scheme  [41 ] to derive the quantum version of a classical game, grounded in Fine's theorem from the early 1980s. Utilizing Positive Operator-Valued Measures (POVMs), we ascertain the quantum states that rectify inherent paradoxes in classical games;
  • Study of quantum games that are played on networks [23 ]. This research explores the impact of quantum strategies on player behaviour in network games, where player interactions are modelled via network topology. Our findings reveal that quantum correlations can significantly influence the equilibrium strategies and outcomes of games that are played on networks;
  • My work also extends to quantum Bayesian games [24 25 ], which possess a more intricate underlying probability structure and offer a rich backdrop to examine the role of quantum probabilities. These papers open a new pathway for applying quantum games to analyse decision-making and information exchange in games characterized by incomplete information;
  • Investigating [26 ] the application of quantum games to improve our understanding and portrayal of concept combinations in human cognition.

Game Theory Applications: My notable contributions in this field encompass:

  • The development of an extension to Selten's widely recognized ransom kidnapping game model [27 ];
  • An insightful review article [40 ] discussing the applications of game theory in the domain of network and cybersecurity.

Geometric Algebra Applications: Geometric Algebra (GA) marries the algebraic structure of Clifford’s algebra with a discernible geometric interpretation, thus enhancing geometric intuition with the precision of an algebraic system. My work on applying GA includes:

  • A GA-driven analysis of Meyer's quantum penny-flip game [28 ];
  • A study of two-player and three-player quantum games in an EPR-type setup, leveraging GA [29 30 ];
  • Exploration of special relativity through the mathematical formalism of GA [31 ];
  • An investigation of N-player quantum games within an EPR context [32 ];
  • Advancement of an enhanced formalism for quantum computation grounded in GA and its implementation in Grover's search algorithm [33 ];
  • An investigation into the advantages of GA formalism for engineers [34 ];
  • A study of the functions of multivector variables within GA [35 ];
  • Examination of time as a geometric property within the GA-conceived perception of space [36 ];
  • Showing how the structure of Minkowski's four dimensional spacetime continuum emerges as a natural property of physical three-dimensional space, if it is modeled with GA [43].

Mathematical Modelling: This field involves the use of mathematical concepts and language for system descriptions. My contributions include:

  • Formulation of mathematical models for memristive devices, and their applications in circuits and system simulations [37 ,38 ,39 ]. Memristors, a class of passive circuit element, can store information and maintain a relationship between the time integrals of current and voltage across a two-terminal element;
  • Supervision of a research project concentrating on the mathematical modeling of the COVID-19 outbreak in Bahrain [41 ,42].

To view my research impact indicators, please visit the following links: ORCID Web of Science Google Scholar Scopus Loop ResearchGate , and Academia .

  • Faculty of Engineering, Computer & Mathematical Sciences (ECMS) Interdisciplinary Research Grant Scheme 2016 (jointly with Prof Derek Abbott & Dr Virginie Masson) at the University of Adelaide, AU$ 30,000 (2016-2017)
  • Discovery Research Grant DP0771453 and Fellowship (Principal Investigator) from Australian Research Council (ARC) at University of Adelaide, AU$ 247,092 (2007-2011)
  • Research Grant P06330 and Fellowship (Principal Investigator) from Japan Society for the Promotion of Science (JSPS) at Kochi University of Technology, Japanese Yen 4,958,500 (2006-2007)
  • Fully funded PhD Research Scholarship from the University of Hull, UK, for overseas research students (2002-2005)
  • Fully funded Merit Scholarship from the Government of Pakistan for studying overseas at the University of Sheffield, UK (1992-1995)

Department of Mathematics, College of Science, University of Bahrain (UoB):

2nd Semester 2020-2021:

  • Fluid Mechanics (Level 3)
  • Calculus II (Level 1)
  • Calculus & Analytical Geometry II (Level 1)
  • Calculus & Analytical Geometry III (Level 2)

1st Semester 2020-2021:

  • Analytical Mechanics (Level 3)
  • Methods of Applied Mathematics (Level 3)
  • Calculus II (Level 1)

2nd Semester 2019-2020:

  • Calculus II (Level 1)
  • Maths for Business Management (Level 1)
  • Calculus & Analytic Geometry III (Level 2)

School of Electrical & Electronic Engineering, University of Adelaide:

  • Avionic Sensors & Systems Combined (Level 4), 2014 Semester 2: Guest Lecturer
  • Communications/Principles of Communication Systems (Combined) (Level 4), 2012 Semester 1: Guest Lecturer
  • Communications/Principles of Communication Systems (Combined) (Level 4), 2011 Semester 1: Guest Lecturer

Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals (KFUPM):

  • Methods of Applied Mathematics (Level 3), Jan 2013 to May 2014, taught this course 4 times
  • Elements of Differential Equations (Level 2), Jan 2013 to May 2014, taught this course twice

School of Natural Sciences, National University of Sciences & Technology (NUST):

  • Mathematical Foundations of Quantum Mechanics (Level 4), July-Nov 2006

Riphah International University (RUI):

  • Engineering Electromagnetics (Level 2), Sep 2000-Sep 2001

Tutoring experience

School of Electrical & Electronic Engineering, University of Adelaide:

  • Vector Calculus & Electromagnetics (Level 2), 2022 Semester 2
  • Electronic Circuits (Level 2), 2022 Semester 1
  • Electronic Circuits (Level 2), 2018 Semester 1
  • Electronic Circuits (Level 2), 2017 Semester 1
  • Electronic Circuits (Level 2), 2016 Semester 1
  • Electronic Systems (Level 1), 2016 Semester 1

Maths Learning Centre (MLC), University of Adelaide:

  • Undergrad Maths courses (Various Levels), 2017 to 2019
  • Position: Research Fellow (A) (with PhD)
  • Phone: 83135589
  • Email:
  • Campus: North Terrace
  • Building: Ingkarni Wardli, floor Level Three
  • Org Unit: Electrical and Electronic Engineering

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