A quantum algorithm for advection-diffusion equation by a probabilistic imaginary-time evolution operator

In this work, we propose a new quantum algorithm for solving the linear advection-diffusion reaction equations by employing a novel approximate probabilistic imaginary-time evolution (PITE) operator. This algorithm based on the approximate PITE operator can be easily extended to evolution equations. In more detail, we first verify the effectiveness of the proposed approximate PITE operator by the theoretical evaluation of the error. Next, we construct the explicit quantum circuit for realizing the imaginary-time evolution of the Hamiltonian coming from the advection-diffusion equation, whose gate complexity is logarithmic regarding the size of the discretized Hamiltonian matrix. To verify our algorithm, numerical simulations using gate-based quantum emulator Qiskit for 1D/2D examples are also provided. Moreover, we compare our proposed algorithm to some other previous works and find that our algorithm gives comparable result to the Harrow-Hassidim-Lloyd (HHL) algorithm with similar gate complexity, although we use much less ancillary qubits. Besides, our algorithm outperforms a specific HHL algorithm and a variational quantum algorithm based on the finite difference method. Finally, we extend our algorithm to the coupled system of advection-diffusion reaction equations, and we also demonstrate some simulation results for nonlinear reaction-diffusion systems, including Burgers' equation, provided that time-step-wise measurements are allowed.