Resource-efficient quantum simulation of transport phenomena via Hamiltonian embedding

Simulation of high-dimensional transport PDEs is classically intractable. While quantum algorithms offer exponential speedups, existing methods are often heuristic or require fault-tolerant hardware with complex input models like QRAM, making them infeasible for near-term devices. In this work, we develop a resource-efficient, end-to-end framework using Hamiltonian embedding for quantum PDE solvers. This "white-box" approach maps the PDE to a sparse Hamiltonian via discretization and Schrödingerization. This sparse Hamiltonian is then embedded into a larger, hardware-efficient Hamiltonian of local Pauli operators, which is compiled directly to elementary gates. This method circumvents complex input models and achieves a provable exponential speedup, with gate complexity scaling nearly-linearly in time and polylogarithmically in error when combined with Richardson extrapolation. We demonstrate this framework for linear advection and nonlinear scalar hyperbolic equations. Resource analyses show sparse encodings (one-hot, unary) outperform standard binary encoding. Finally, we report the first real-machine simulation of a 2D advection equation, implemented on the IonQ Aria-1 quantum processor.