A novel quantum algorithm for solving advection–diffusion equation by the lattice Boltzmann method is proposed. The presented quantum algorithm is composed of two major segments. In the first segment, equilibrium distribution function, presented in the form of a non-unitary diagonal matrix, is quantum circuit implemented by using a standard-form encoding approach. For the second segment, the quantum walk procedure as a method for implementing the propagation step is applied. The constructed algorithm is presented as a series of single- and two-qubit gates, as well as multiple-input controlled-NOT gates. In order to demonstrate the validity of the proposed quantum algorithm, the unsteady one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations are solved by using the IBM’s quantum computing software development framework Qiskit, while the analytic solution and the classic code are used for verification. Finally, the complexity analysis and directions for future work are discussed.

提出了一种用格子玻尔兹曼方法求解对流扩散方程的新型量子算法。提出的量子算法由两个主要部分组成。在第一部分中，以非-对角矩阵形式表示的平衡分布函数是通过使用标准形式的编码方法实现的量子电路。对于第二段，应用量子行走程序作为实现传播步骤的方法。所构造的算法以一系列单比特和双量子比特门以及多输入受控NOT门的形式呈现。为了证明所提出的量子算法的有效性，使用IBM的量子计算软件开发框架Qiskit求解了不稳定的一维（1D）和二维（2D）对流扩散方程，同时使用解析解决方案和经典代码进行验证。最后，讨论了复杂性分析和未来工作的方向。