The advection-diffusion equation is simulated on a superconducting quantum computer via several quantum algorithms. Three formulations are considered: (1) Trotterization, (2) variational quantum time evolution (VarQTE), and (3) adaptive variational quantum dynamics simulation (AVQDS). These schemes were originally developed for the Hamiltonian simulation of many-body quantum systems. The finite-difference discretized operator of the transport equation is formulated as a Hamiltonian and solved without the need for ancillary qubits. Computations are conducted on a quantum simulator (IBM Qiskit Aer) and an actual quantum hardware (IBM Fez). The former emulates the latter without the noise. The predicted results are compared with direct numerical simulation (DNS) data with infidelities of the order $10^{-5}$. In the quantum simulator, Trotterization is observed to have the lowest infidelity and is suitable for fault-tolerant computation. The AVQDS algorithm requires the lowest gate count and the lowest circuit depth. The VarQTE algorithm is the next best in terms of gate counts, but the number of its optimization variables is directly proportional to the number of qubits. Due to current hardware limitations, Trotterization cannot be implemented, as it has an overwhelming large number of operations. Meanwhile, AVQDS and VarQTE can be executed, but suffer from large errors due to significant hardware noise. These algorithms present a new paradigm for computational transport phenomena on quantum computers.

翻译：本文通过多种量子算法在超导量子计算机上模拟了平流-扩散方程。考虑了三种实现方案：(1) 特罗特分解法，(2) 变分量子时间演化算法，以及 (3) 自适应变分量子动力学模拟算法。这些方案最初是为多体量子系统的哈密顿量模拟而开发的。将输运方程的有限差分离散化算子表述为哈密顿量，并在无需辅助量子比特的情况下求解。计算在量子模拟器（IBM Qiskit Aer）和实际量子硬件（IBM Fez）上进行。前者在无噪声条件下模拟后者。预测结果与直接数值模拟数据进行了比较，保真度误差约为 $10^{-5}$ 量级。在量子模拟器中，观察到特罗特分解法具有最低的保真度误差，适用于容错计算。自适应变分量子动力学模拟算法所需的量子门数量和电路深度最低。变分量子时间演化算法在量子门数量方面次优，但其优化变量的数量与量子比特数成正比。由于当前硬件限制，特罗特分解法因操作数量极其庞大而无法实现。同时，自适应变分量子动力学模拟算法和变分量子时间演化算法虽可执行，但因显著的硬件噪声而存在较大误差。这些算法为量子计算机上的计算输运现象研究提供了新范式。