Modeling Heat Conduction with an Internal Heat Source Using a Quantum Lattice Boltzmann Method

This work presents a fully quantum implementation of the Quantum Lattice Boltzmann Method (QLBM) for simulating one-dimensional heat conduction with internal heat generation. Building on the classical Lattice Boltzmann framework, QLBM reformulates the collision and streaming steps as unitary operations on quantum circuits, enabling site-wise parallel updates on near-term quantum hardware. In this formulation, classical distribution functions are encoded into quantum amplitudes, and the evolution of the system is performed entirely through quantum gates, including a collision operator and a streaming step realized as a qubit permutation. The model is validated by simulating a canonical diffusion problem with a localized heat source, and results are benchmarked against both classical LBM and an analytical solution obtained via ODE integration. The impact of quantum sampling noise is systematically assessed by varying the number of measurement shots, revealing that higher shot counts significantly improve fidelity to classical results. Root-mean-square error analyses confirm that the fully quantum model accurately captures the spatiotemporal behavior of the system, demonstrating the viability of QLBM for modeling non-unitary physics in thermally driven systems.