Hybrid classical-quantum computation of heat diffusion in multilayer materials

Solving the 1D time-dependent heat diffusion equation using the Finite Difference Time Domain method is simplified to multiplying a tridiagonal sparse matrix by a vector representing the heat (temperature) distribution. To implement this multiplication operation with a quantum algorithm, we design and describe a quantum circuit for matrix-vector multiplication. The sparse matrix is loaded using recursive divide-and-conquer approach with binary tree preorder traversal. Subsequently, heat diffusion is simulated for a two-layer medium with different diffusion coefficients, and its distribution across the layers is tracked over time. The critical time is approximated based on the distribution of diffused heat. The critical time from the numerical simulation was validated using two empirical methods. It matches well with the more complex method, while the simpler method yielded a different value.