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Implicit methods are attractive for hybrid quantum-classical CFD solvers asthe flow equations are combined into a single coupled matrix that is solved onthe quantum device, leaving only the CFD discretisation and matrix assembly onthe classical device. In this paper, an implicit hybrid solver is investigatedusing emulated HHL circuits. The hybrid solutions are compared with classicalsolutions including full eigen-system decompositions. A thorough analysis ismade of how the number of qubits in the HHL eigenvalue inversion circuit affectthe CFD solver's convergence rates. Loss of precision in the minimum andmaximum eigenvalues have different effects and are understood by relating thecorresponding eigenvectors to error waves in the CFD solver. An iterativefeed-forward mechanism is identified that allows loss of precision in the HHLcircuit to amplify the associated error waves. These results will be relevantto early fault tolerant CFD applications where every (logical) qubit willcount. The importance of good classical estimators for the minimum and maximumeigenvalues is also relevant to the calculation of condition number for QuantumSingular Value Transformation approaches to matrix inversion.

Implicit Hybrid Quantum-Classical CFD Calculations using the HHL Algorithm