Comparison of Mathematical Models for Subscription Services Using Optimization Problems and Quantum Information Theory -Feasibility of Implementing Optimization Problem Algorithms on Quantum Computers-
Misao Fukuda
physics.soc-phquant-phSubmitted Nov 16, 2025• Announced Nov 18, 2025• 0 citations
The purpose of this research is to explore whether it is possible to construct a design theory for subscription services for intangible goods from a time discounting perspective, based on quantum information theory, which is the foundational theory for quantum computers and similar technologies. To this end, we propose a mathematical model of subscription services using optimization problems based on optimal growth theory from standard economics, and with reference to microeconomics, we define utility as a value function of customer satisfaction derived from quantum mutual information, an entropy measure in quantum information theory, by considering time discounting. We propose the quantification of customer satisfaction and the formulation of consumer surplus. In the mathematical model of subscription services, the existence of a minimum value in the time-discounted customer satisfaction value function under budget constraints, and the realization of a mathematical expression for consumer surplus, could be explained by the laws of behavioral economics. This yielded new insights into the design of individually customized customer experiences, enhanced the feasibility of constructing economic models based on quantum information theory and the mathematical design of customer experiences, raised the possibility that mathematical models using quantum information theory can achieve greater economic welfare than standard economics, and increased the feasibility of implementing optimization problem algorithms on quantum computers.
The purpose of this research is to explore whether it is possible to construct a design theory for subscription services for intangible goods from a time discounting perspective, based on quantum information theory, which is the foundational theory for quantum computers and similar technologies. To this end, we propose a mathematical model of subscription services using optimization problems based on optimal growth theory from standard economics, and with reference to microeconomics, we define utility as a value function of customer satisfaction derived from quantum mutual information, an entropy measure in quantum information theory, by considering time discounting. We propose the quantification of customer satisfaction and the formulation of consumer surplus. In the mathematical model of subscription services, the existence of a minimum value in the time-discounted customer satisfaction value function under budget constraints, and the realization of a mathematical expression for consumer surplus, could be explained by the laws of behavioral economics. This yielded new insights into the design of individually customized customer experiences, enhanced the feasibility of constructing economic models based on quantum information theory and the mathematical design of customer experiences, raised the possibility that mathematical models using quantum information theory can achieve greater economic welfare than standard economics, and increased the feasibility of implementing optimization problem algorithms on quantum computers.
Abstract Quantum computing is increasingly discussed as a possible source of new computational methods for economics, but its relevance depends on whether economic bottlenecks match quantum algorithmic primitives. This critical review evaluates that match for computational economics. It synthesizes work on quantum amplitude estimation and quantum Monte Carlo for simulation and risk measurement; quadratic unconstrained binary optimization (QUBO), Ising formulations, quantum annealing, and the quantum approximate optimization algorithm for discrete optimization; and quantum linear-algebra and machine-learning primitives for econometrics and model solution. The review argues that there is no general-purpose quantum algorithm for economics analogous to Shor's algorithm for factoring. Credible claims are problem-class specific and depend on state preparation, output requirements, conditioning, hardware noise, embedding overhead, and comparison with strong classical and quantum-inspired baselines. The most mature economic applications are derivative pricing and financial risk, payment-system liquidity optimization, quantum Monte Carlo for stress testing, and early quantum-annealing formulations of dynamic programming. Transportation and infrastructure applications are relevant mainly as networked economic optimization problems. The article proposes an evidence ladder for quantum computational economics and a research agenda centered on reproducible benchmarks, economically meaningful objective functions, end-to-end resource accounting, and evaluation by economic value rather than hardware novelty. JEL Classification: C02; C15; C61; C63; C88; E37; G11; G17; R41
Introduction. Quantum computers provide several times faster solutions to several NP-hard combinatorial optimization problems in comparison with computing clusters. The trend of doubling the number of qubits of quantum computers every year suggests the existence of an analog of Moore's law for quantum computers, which means that soon they will also be able to get a significant acceleration of solving many applied large-scale problems. The purpose of the article is to review methods for creating algorithms of quantum computer mathematics for combinatorial optimization problems and to analyze the influence of the qubit-to-qubit coupling and connections strength on the performance of quantum data processing. Results. The article offers approaches to the classification of algorithms for solving these problems from the perspective of quantum computer mathematics. It is shown that the number and strength of connections between qubits affect the dimensionality of problems solved by algorithms of quantum computer mathematics. It is proposed to consider two approaches to calculating combinatorial optimization problems on quantum computers: universal, using quantum gates, and specialized, based on a parameterization of physical processes. Examples of constructing a half-adder for two qubits of an IBM quantum processor and an example of solving the problem of finding the maximum independent set for the IBM and D-wave quantum computers are given. Conclusions. Today, quantum computers are available online through cloud services for research and commercial use. At present, quantum processors do not have enough qubits to replace semiconductor computers in universal computing. The search for a solution to a combinatorial optimization problem is performed by achieving the minimum energy of the system of coupled qubits, on which the task is mapped, and the data are the initial conditions. Approaches to solving combinatorial optimization problems on quantum computers are considered and the results of solving the problem of finding the maximum independent set on the IBM and D-wave quantum computers are given. Keywords: quantum computer, quantum computer mathematics, qubit, maximal independent set for a graph.
This article explores the interdisciplinary interface between quantum computing and economic theory, with a particular emphasis on how foundational concepts from quantum mechanics reshape core methodological and epistemological assumptions in economics. Drawing on a comprehensive bibliographic methodology, the study synthesizes contributions from quantum physics, computational theory, and economic modeling to trace the impact of quantum logic -especially principles such as superposition, entanglement, and measurement irreversibility- on both theoretical constructs and applied frameworks. Particular attention is given to emblematic algorithms (e.g., Shor’s and Grover’s) as case studies for illustrating computational discontinuities in fields such as cryptographic trust, portfolio optimization, and decision-making under uncertainty. The article also surveys recent developments in quantum machine learning and quantum-inspired models relevant to economic forecasting and adaptive policy design. Ethical and distributional concerns are critically addressed, especially in relation to the asymmetric global access to quantum resources. By mapping this emerging field through a structured review of cross-disciplinary literature, the paper offers a conceptual framework for understanding how quantum technologies may influence future research in financial economics, risk theory, and the modeling of complex systems operating under deep informational constraints.
Quantum Artificial Intelligence has been considered as one of transformative interdisciplinary areas that have joined quantum calculation with machine learning and optimization. The paper is a survey of the recent development in QAI and a detailed case study of the Quantum Approximate Optimization Algorithm, its mathematical formulation, operator-based modeling, and utility in practical application to combinatorial optimization. Based on the background principles of the Hilbert space theory, Hermitian operators and variational quantum circuits, the paper examines the process of encoding a classical optimization problem in a quantum Hamiltonian and the effects of circuit depth, parameterization, and graph structure on the algorithms. An example of Max-Cut case study shows how the algorithm performs on regular graph, irregular graph and weighted graph cases, showing good approximation even at small depths, and showing convergence behavior as outlined in theory. The comparative perspectives in regards to existing models of QAI further outline the advantages of QAOA with regard to interpretability and operator design, and also outline the constraints associated with noise sensitivity, measurement load, and hardware scalability. The way QAOA is related to spectral methods, variational theory, and discrete optimization are discussed as giving theoretical implications of QAOA to applied mathematics. The research finds that QAOA is a bright future of practical, yet mathematically rich, quantum computation in the near-term. The development in the future depends on the interdisciplinary cooperation, noise-resistant circuit engineering, and the development of quantum hardware to achieve the full potential of QAI in both science and engineering uses.
Abstract Microeconomics represents the most formally formalized branch of the social sciences. Yet its core actor—the rational agent endowed with complete and consistent preferences, who maximizes a well-defined utility function without hesitation—does not exist in laboratory experiments, market transactions or real household decision-making. Behavioral economics has dismantled this myth, only to replace it with a disjoint catalogue of separate “effects” and “anomalies”: different effects yield contradictory predictions, and researchers select effects ex post to rationalize all observed outcomes. A discipline that explains everything case by case can barely generate ex ante predictions. This monograph is the first work worldwide to systematically reconstruct quantum microeconomics. Human preferences and beliefs possess a mathematical structure isomorphic to quantum states: prior to a decision, alternative options coexist in superposition; choice constitutes a measurement that constructs preferences; question order alters responses because judgments correspond to incompatible observables; interacting individuals become mutually entangled. Framing effects, order effects, preference reversals and herding behavior are no longer a mere inventory of biases, but predictable signatures emerging from a unified underlying process. This book does not argue that the human brain operates as a physical quantum machine. It posits instead that the mathematical structure of human choice is isomorphic to quantum statesarXiv. The book is organized into six parts. Part I, Paradigm Revolution, diagnoses the century-long crisis of the classical paradigm, traces its mechanistic legacy from Newton to Walras, establishes the philosophical foundations of practical ontology and observation dependence, puts forward the quantum agent hypothesis, and assembles the analytical framework for the entire volume. Part II develops quantum consumer theory, reconstructing preferences, utility, choice, demand and welfare measurement on the basis of superposition. Part III builds quantum producer theory, covering production, costs, supply, the theory of the firm, and profit maximization. Part IV addresses quantum market structure and game theory, treating perfect competition, monopoly, monopolistic competition, oligopoly and games as entangled quantum systems. Part V reconstructs quantum general equilibrium and market failure, including general equilibrium theory, welfare economics, externalities, public goods and asymmetric information. Part VI extends the framework to factor markets for labor, capital and land, synthesizes behavioral microeconomics, and charts future disciplinary frontiers. The book balances unity and depth, originality and rigor, formal rigor and readability. Within an axiomatic framework, it derives core macroeconomic phenomena from a small set of quantum principles. Each chapter contains original theoretical construction, formal mathematical derivation and real-world applications, with purpose-built diagrams to substantiate arguments. This volume is suitable for undergraduate and graduate students, academic researchers, policy practitioners, and all readers who seek to understand why economies boom and collapse.
Optimization is a crucial challenge across various domains, including finance, resource allocation, and mobility. Quantum computing has the potential to redefine the way we handle complex problems by reducing computational complexity and enhancing solution quality. Optimization, particularly of objective functions, stands to benefit significantly from quantum solvers, which leverage principles of quantum mechanics like superposition, entanglement, and tunneling. The Ising and Quadratic Unconstrained Binary Optimization (QUBO) models are the most suitable formulations for these solvers, involving binary variables and constraints treated as penalties within the overall objective function. To harness quantum approaches for optimization, two primary strategies are employed: exploiting quantum annealers—special-purpose optimization devices—and designing algorithms based on quantum circuits. This review provides a comprehensive overview of quantum optimization methods, examining their advantages, challenges, and limitations. It demonstrates their application to real-world scenarios and outlines the steps to convert generic optimization problems into quantum-compliant models. Lastly, it discusses available tools and frameworks that facilitate the exploration of quantum solutions for optimization tasks.