Huang et al. at Quemix have proposed a new quantum algorithm using the probabilistic imaginary-time evolution (PITEⓇ) method, achieving a high-precision solution to the linear advection-diffusion equation. This algorithm constructs a quantum circuit with logarithmic gate complexity and demonstrates its effectiveness through numerical simulations in one and two dimensions. Furthermore, compared to the Harrow-Hassidim-Lloyd (HHL) algorithm and variational quantum algorithms (VQA) based on the finite difference method (FDM), it is emphasized that this algorithm achieves comparable results while requiring fewer ancillary qubits. Future applications to nonlinear systems are anticipated.
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