Schrodingerization based quantum algorithms for the time-fractional heat equation

We develop a quantum algorithm for solving high-dimensional time-fractional heat equations. By applying the dimension extension technique from FKW23, the d+1-dimensional time-fractional equation is reformulated as a local partial differential equation in d+2 dimensions. Through discretization along both the extended and spatial domains, a stable system of ordinary differential equations is obtained by a simple change of variables. We propose a quantum algorithm for the resulting semi-discrete problem using the Schrodingerization approach from JLY24a, JLY23, JL24a. The Schrodingerization technique transforms general linear partial and ordinary differential equations into Schrodinger-type systems--with unitary evolution, making them suitable for quantum simulation. This is accomplished via the warped phase transformation, which maps the equation into a higher-dimensional space. We provide detailed implementations of this method and conduct a comprehensive complexity analysis, demonstrating up to exponential advantage--with respect to the inverse of the mesh size in high dimensions~--~compared to its classical counterparts. Specifically, to compute the solution to time T, while the classical method requires at least O (Nₜ d h^- (d+0. 5) ) matrix-vector multiplications, where Nₜ is the number of time steps (which is, for example, O (Tdh^-2) for the forward Euler method), our quantum algorithms requires O (T²d⁴ h^-8) queries to the block-encoding input models, with the quantum complexity being independent of the dimension d in terms of the inverse mesh size h^-1. Numerical experiments are performed to validate our formulation.