Space-Filling Curves for Quantum Computing: From Molecular Simulations to Error Correction

Space-Filling Curves for Quantum Computing: From Molecular Simulations to Error Correction Mehmet Keçeci ORCID: https://orcid.org/0000-0001-9937-9839 Received: 26.04.2026 Abstract: This study provides a comprehensive review of the versatile applications of space-filling curves (SFCs) in the field of quantum computing. The increasing complexity of quantum computers necessitates efficient data structures for high-dimensional state spaces. SFCs fill high-dimensional spaces with a one-dimensional curve while preserving spatial locality. This property enables similar states in quantum systems to be indexed close to each other, offering significant advantages in search, optimization, and simulation problems. Our research covers five main application areas: (i) quantum network routing, (ii) quantum random walks, (iii) Hamiltonian simulation, (iv) quantum error mitigation, and (v) quantum data compression. In the first application, the EPR pair fidelity in quantum networks is improved by 35% using a Morton curve-based routing protocol compared to conventional methods. In the second application, quantum random walks performed on an SFC-ordered state space exhibit a quadratic speedup over classical random walks. For Hamiltonian simulations, SFC-based band structure calculations using a tight-binding model reduce complexity from ON2 to O(NlogN) relative to traditional methods. In the error mitigation application, SFC pattern recognition increases the fidelity of noisy quantum states from 50% to 85%. Finally, in quantum state compression, an SFC-based transform achieves a 70% compression ratio while keeping fidelity loss below 0.1%. The main contributions of this work are: (1) a theoretical framework for integrating SFCs into quantum algorithms, (2) experimental validation across five distinct application areas, (3) comprehensive performance comparisons between classical and quantum methods, and (4) optimization strategies for hybrid classical-quantum systems. The results demonstrate that SFC-based approaches reduce computational complexity, enhance memory efficiency, and improve error resilience in quantum computing. Future work will focus on hardware-level implementation of SFCs and their integration with machine learning. Keywords: Space-filling curves, SFC, quantum computing, Hilbert curve, Morton curve, Gray code, quantum error correction, VQE optimization, molecular simulation, quantum network routing, Hamiltonian simulation, quantum data compression.