Implicit Hybrid Quantum-Classical CFD Calculations using the HHL Algorithm

Abstract

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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“…The HHL quantum algorithm for linear systems of equations, named after its authors Harrow, Hassidim, and Lloyd [78], and variations thereof, thus have the potential to speed up CFD simulations. As noted by Lapworth [79], classical algorithms running on supercomputers are highly efficient at solving matrix equations by, for example, side-stepping the need for matrix inversions. Quantum algorithms do not need to, even should not rely on the exact approximations as classical algorithms, however.…”

“…Quantum algorithms like HHL and the Quantum Singular Value Transformation (QSVT) [80] can efficiently perform direct matrix inversions and should, therefore, be utilized for quantum advantage. The approach presented by Lapworth [79] relies on fault-tolerant quantum computers, but hybrid classical/quantum algorithms for the NISQ era have been proposed and discussed [81].…”