A Quantum Search Approach to Magic Square Constraint Problems with Classical Benchmarking

2026-04-06

AI Summary

This work addresses the magic square construction problem within combinatorial constraint satisfaction by proposing a non-iterative classical-quantum协同 framework. The classical component employs the Siamese construction method together with partial constraint checking to perform structured preprocessing, thereby reducing the solution space. The quantum component formulates the problem as a Grover search task, featuring a reversible and constraint-aware oracle implemented via multi-register modular arithmetic circuits to enable amplitude amplification. Experiments conducted in Qiskit demonstrate the correctness and efficacy of the approach on small-scale magic square instances, achieving a quadratic speedup in query complexity compared to classical brute-force enumeration and backtracking algorithms, thus highlighting the potential of quantum computing for such combinatorial problems.

Technology Category

Machine Learning: Quantum Machine LearningSearch and Optimization: Combinatorial OptimizationConstraint Satisfaction and Optimization: Satisfiability

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSecurity and Privacy: Applications of cryptographyEconomics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

This paper presents a quantum search approach to combinatorial constraint satisfaction problems, demonstrated through the generation of magic squares. We reformulate magic square construction as a quantum search problem in which a reversible, constraint-sensitive oracle marks valid configurations for amplitude amplification via Grover's algorithm. Classical pre-processing using the Siamese construction and partial constraint checks generates a compact candidate domain before quantum encoding. Rather than integrating classical and quantum solvers in an iterative loop, this work uses the classical component for structured initialisation and the quantum component for search, and benchmarks the quantum approach against classical brute-force enumeration and backtracking. Our Qiskit implementation demonstrates the design of multi-register modular arithmetic circuits, oracle logic, and diffusion operators. Experiments are conducted on small grid instances, as larger grids are intractable on classical statevector simulators due to exponential memory growth. The results validate the correctness of the proposed quantum search pipeline and confirm the theoretical quadratic query advantage over classical search.

Problem

• quantum search
• magic square
• constraint satisfaction problem
• Grover's algorithm
• combinatorial optimization

Innovation

• quantum search
• constraint satisfaction
• Grover's algorithm
• magic square
• oracle design