Quantum time-marching algorithm for solving the bounded Diffusion equation
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We present the implementation of a time-marching algorithm for solving diffusion problems on fault-tolerant quantum computers. The proposed method is capable of handling a wide variety of boundary conditions, retaining optimal success probabilities without additional amplification techniques and maintaining linear time complexity. For the treatment of non-unitary diffusive dynamics, the linear combination of unitaries algorithm implements the unitarily decomposition of the discrete operator. To incorporate boundary conditions, the method of images uses adjacent discrete points by relating the boundary type to the symmetry of the non-periodic dimension. Results of the two-dimensional state-vector simulations demonstrate excellent predictive agreement with classical finite differences simulations, conserving the optimal problem-intrinsic success probabilities. The approach can be generalized to arbitrary dimensions.
