Quantum time-marching algorithms for solving linear transport problems including boundary conditions

This article presents the first complete application of a quantum time-marching algorithm for simulating multidimensional linear transport phenomena with arbitrary boundaries, whereby the success probabilities are problem intrinsic. The method adapts the linear combination of unitaries algorithm to block encode the diffusive dynamics, while arbitrary boundary conditions are enforced by the method of images only at the cost of one additional qubit per spatial dimension. As an alternative to the non-periodic reflection, the direct encoding of Neumann conditions by the unitary decomposition of the discrete time-marching operator is proposed. All presented algorithms indicate optimal success probabilities while maintaining linear time complexity, thereby securing the practical applicability of the quantum algorithm on fault-tolerant quantum computers. The proposed time-marching method is demonstrated through state-vector simulations of the heat equation in combination with Neumann, Dirichlet, and mixed boundary conditions.