使用HHL算法的隐式混合量子-经典CFD计算
Implicit Hybrid Quantum-Classical CFD Calculations using the HHL Algorithm
摘要：
隐式方法对于混合量子-经典CFD求解器是有吸引力的，因为流动方程被组合成在量子设备上求解的单个耦合矩阵，仅留下经典设备上的CFD离散化和矩阵组装。本文利用仿真HHL电路研究了隐式混合求解器。混合解决方案与经典的解决方案，包括充分的本征系统分解。深入分析了HHL本征值反演电路中的量子位数对计算流体力学求解器收敛速度的影响。最小和最大特征值的精度损失具有不同的影响，并通过将相应的特征向量与CFD求解器中的误差波相关联来理解。一个迭代的前馈机制被确定，允许在HHL电路的精度损失放大相关的误差波。这些结果将与早期容错CFD应用相关，其中每个（逻辑）量子位都将计数。最小和最大特征值的好的经典估计的重要性也与量子奇异值变换方法的条件数的计算有关。
英文摘要：
Implicit methods are attractive for hybrid quantum-classical CFD solvers as the flow equations are combined into a single coupled matrix that is solved on the quantum device, leaving only the CFD discretisation and matrix assembly on the classical device. In this paper, an implicit hybrid solver is investigated using emulated HHL circuits. The hybrid solutions are compared with classical solutions including full eigen-system decompositions. A thorough analysis is made of how the number of qubits in the HHL eigenvalue inversion circuit affect the CFD solver's convergence rates. Loss of precision in the minimum and maximum eigenvalues have different effects and are understood by relating the corresponding eigenvectors to error waves in the CFD solver. An iterative feed-forward mechanism is identified that allows loss of precision in the HHL circuit to amplify the associated error waves. These results will be relevant to early fault tolerant CFD applications where every (logical) qubit will count. The importance of good classical estimators for the minimum and maximum eigenvalues is also relevant to the calculation of condition number for Quantum Singular Value Transformation approaches to matrix inversion.
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