A quantum algorithm for heat conduction with symmetrization

Heat conduction, driven by thermal non-equilibrium, is the transfer of internal thermal energy through physical contacts, and it exists widely in various engineering problems, such as spacecraft and state-of-the-art dilution refrigerators. The mathematical equation for heat conduction is a prototypical partial differential equation. Here we report a quantum algorithm for heat conduction (QHC) that significantly outperforms classical algorithms. We represent the original heat conduction system by a symmetric system with an ancilla qubit so that the quantum circuit complexity is polylogarithmic in the number of discretized grid points. Compared with the existing algorithms based on solving linear equations via the Harrow-Hassidim-Lloyd (HHL) algorithm, our method evolves the linear process directly without phase estimation, which involves complex quantum operations and large output error. Therefore, this algorithm is experimental-friendly and without output error after the discretization procedure. We experimentally implemented the algorithm for a one-dimensional thermal conduction process with two-edge constant temperatures and adiabatic conditions on a nuclear spin quantum processor. The spatial and temporal distributions of the temperature are accurately determined from the experimental results. Our work can be naturally applied to any physical processes that can be reduced to the heat equation.