Abstract:
Game theory provides a mathematical framework for the study of strategic interactions among rational agents whose decisions influence one another. Concepts like best responses, Nash equilibrium and Pareto optimality have been widely used to study decision-making processes in economics and in computer science as well. At the same time, many problems of practical interest can be formulated into a classical optimization problem, which is generally computationally difficult. Recent developments in variational quantum algorithms (VQA), have led to a new technique called Quantum Approximate Optimization Algorithm (QAOA), which aims to address certain classical optimization problems using hybrid quantum–classical approaches. In this work, we introduce a quantum system in which ideas from game theory are applied for the optimization of quantum systems. We formulate a quantum game involving two agents acting on a shared multi-qubit system, where each player seeks to optimize the expectation value of a given payoff operator. We analyze the behavior of this system through numerical simulations and investigate the emergence of equilibrium-like configurations in the parameter space of quantum states. This perspective connects concepts from game theory, quantum optimization, and variational quantum algorithms. The study may also provide insights into the design of multi-agent reinforcement learning algorithms for quantum-mechanical systems, where multiple learning agents interact through a shared quantum environment.
