一种基于对称化热传导系统的量子算法

A quantum algorithm for heat conduction with symmetrization

由非平衡驱动的热传导通过物理接触传递内部的热能.描述热传导的数学方程是一个典型的偏微分方程.它广泛存在于各种工程问题中,例如粒子扩散、航天器设计和稀释制冷机运行.本文提出了一种显著优于经典算法的热传导量子算法,并在量子硬件上演示了热传导过程.作者通过一个辅助量子比特构造的对称系统来表示原始热传导系统,使得量子线路的复杂度减少到离散格点数量的对数多项式量级.与现有的基于Harrow-Hassidim-Lloyd的线性方程算法相比,本文的方法直接演化线性过程,不进行相位估计,避免相位估计涉及复杂的量子门操作和较大的输出误差,便于在目前的量子硬件实现,显示了该算法是实验友好特性.并且,作者在核自旋量子处理器上实现了两端恒温和绝热条件的一维热传导过程,温度的时空分布与理论预言很好的吻合.该算法可以自然地应用于可归约为热方程的物理过程.

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-ofthe-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.