A universal theory of high-accuracy surface modeling and its quantum intelligent computing methods

<sec><p indent="0mm">To overcome longstanding issues of error propagation and low computational efficiency in geographical information systems (GIS) and computer-aided design (CAD) systems, high-accuracy surface modeling (HASM) methods were developed through the systematic integration of systems theory, optimization cybernetics, and surface theory. Following nearly two decades of numerical experimentation and empirical investigation, a universal fundamental theorem of surface modeling (UFTSM) was formulated. This theorem provides a general theoretical framework applicable to spatial interpolation, upscaling, downscaling, data fusion, and model–data assimilation across a wide range of disciplines, including Earth surface system science, eco-environmental informatics, medical imaging, and computer-aided design. </sec><sec> The UFTSM was first successfully applied to the simulation of eco-environmental surfaces. Within the conceptual framework of the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services (IPBES), eco-environmental components were classified into three categories: nature (including species diversity, ecosystem structure, and geographical features), nature’s contributions to people (such as food provision, freshwater supply, and environmental pollution remediation), and drivers of natural change (including climate change, land-use change, policies, and regulations). The term eco-environmental surface is used as a unified concept to represent surfaces describing nature, nature’s contributions to people, or drivers of natural change. Numerous studies have demonstrated that intrinsic information (e.g., ground-based observations) and extrinsic information (e.g., satellite observations) provide complementary perspectives, and that neither alone can fully characterize an eco-environmental surface. Instead, such surfaces are governed by the joint influence of intrinsic and extrinsic information, and cannot be adequately understood without considering both. To address the challenge of simulating eco-environmental surfaces through the integration of these two information sources, an iterative differential method for HASM-based machine learning was developed, and a fundamental theorem for eco-environmental surface modeling (FTEEM) was proposed. In addition, a suite of high-efficiency algorithms suitable for classical computing platforms, including a modified conjugate gradient algorithm, a multigrid algorithm, an adaptation algorithm, and adjustment computation, was designed to accelerate eco-environmental surface modeling. Compared with existing surface modeling approaches, HASM-based applications exhibit substantially improved accuracy. However, computational cost remains a major bottleneck for global-scale eco-environmental surface modeling, particularly as spatial resolution becomes increasingly fine. </sec><sec> To address the limitations of classical algorithms and hardware, HASM was reformulated as a large sparse linear system using the Lagrange multiplier method. This system was then implemented on both a real quantum computer and a virtual quantum computing platform by employing two widely used quantum linear solvers: the Harrow-Hassidim-Lloyd (HHL) algorithm and the iterative refinement of the variational quantum linear solver (iVQLS). Based on these approaches, two HASM-oriented quantum linear solvers, HASM-HHL and HASM-iVQLS, were developed, enabling the solution of simulation problems that are intractable for classical computers. The respective advantages and limitations of HASM-HHL and HASM-iVQLS, as quantum machine learning algorithms, were systematically evaluated through simulation experiments conducted on both real and virtual quantum computing platforms. The comparative results highlight the urgent need for a universal tool library that supports both classical and quantum intelligent computing. Moreover, the development of a full-stack quantum input–computing–output system is essential for overcoming the principal challenges currently faced by HASM-based quantum machine learning. </sec>