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

TL;DR

This paper introduces a quantum search method for magic square problems, combining classical pre-processing with quantum amplitude amplification, and benchmarks its performance against classical approaches.

Contribution

It presents a novel quantum search framework for constraint problems, integrating classical pre-processing with quantum algorithms, and demonstrates its effectiveness on small instances.

Findings

Quantum approach confirms quadratic speedup over classical search.

Classical pre-processing reduces candidate space before quantum search.

Experiments validate correctness on small grid instances.

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.