Quantum computing introduces a fundamentally new paradigm for simulating physical systems by leveraging superposition, entanglement, and interference to encode and manipulate information through the dynamics of quantum states. The application of quantum algorithms to simulate problems based on partial differential equations (PDEs) has made some progress, particularly through linear system solving; however, other potentially more promising methods remain comparatively underexplored. Here we investigate the suitability of selected PDEs for direct Hamiltonian mapping, examining when and how these equations can be naturally embedded into quantum dynamics. Our primary focus is on the wave equation, due to its significance for electromagnetic (EM) propagation modeling in communication systems. We further extend the Hamiltonian encoding to the heat and Poisson equations. Together, these equations model propagation, diffusion, and steady-state phenomena central to physics and engineering. We identify quantum-specific constraints that arise when mapping these equations to quantum systems. Energy non-conservation, realistic boundaries, and general inhomogeneous media pose challenges not present in classical numerical schemes, such as Yee’s finite-difference time-domain (FDTD) method. Understanding these limitations is essential for determining realistic application domains and for formulating conditions under which quantum simulation can provide an advantage.
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