A Hybrid Quantum-Classical CFD Methodology with Benchmark HHL Solutions

Abstract

There has been significant progress in the development of quantum algorithms for solving linear systems of equations with a growing body of applications to Computational Fluid Dynamics (CFD) and CFD-like problems. This work extends previous work by developing a non-linear hybrid quantumclassical CFD solver and using it to generate fully converged solutions. The hybrid solver uses the SIMPLE CFD algorithm, which is common in many industrial CFD codes, and applies it to the 2-dimensional lid driven cavity test case. A theme of this work is the classical processing time needed to prepare the quantum circuit with a focus on the decomposition of the CFD matrix into a linear combination of unitaries (LCU). CFD meshes with up to 65 × 65 nodes are considered with the largest producing a LCU containing 32,767 Pauli strings. A new method for rapidly re-computing the coefficients in a LCU is proposed, although this reduces, rather than eliminates, the classical scaling issues. The quantum linear equation solver uses the Harrow, Hassidim, Lloyd (HHL) algorithm via a state-vector emulator. Test matrices are sampled from the classical CFD solver to investigate the solution accuracy that can be achieved with HHL. For the smallest 5 × 5 and 9 × 9 CFD meshes, full non-linear hybrid CFD calculations are performed. The impacts of approximating the LCU and the varying the number of ancilla rotations in the eigenvalue inversion circuit are studied. Preliminary timing results indicate that the classical computer preparation time needed for a hybrid solver is just as important to the achievement of quantum advantage in CFD as the time on the quantum computer. The reported HHL solutions and LCU decompositions provide a benchmark for future research. The CFD test matrices used in this study are available upon request.

Implicit Hybrid Quantum-Classical CFD Calculations using the HHL Algorithm

“…Also of interest in Table 1 is that the condition numbers are significantly lower than for the pressure correction matrices reported by [13]. For example, the condition number for the pressure correction matrix on the 17x17 mesh is 3,500 compared to 482 for the implicit matrix.…”

“…As in [13], the steady 2-dimensional incompressible Navier-Stokes equations are considered, which can be written in component form as: Using the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm [14], [32] leads to a set of segregated matrix equations of the form [33]:…”

“…These schemes are sometimes called coupled as the Navier-Stokes equations are solved as a single system. Equation (3) uses the same fixed point Picard linearisation as used by [13], but other linearisations are possible.…”

“…The 8x8 Toeplitz matrix studied by [12] needed only two real numbers to define all the non-zero entries. In previous work [13], the author considered a real world CFD test case and studied the Poisson type pressure correction matrix common to many CFD solvers that use the SIMPLE algorithm and its derivatives [14,15]. Matrices with up to 288 entries, of which 176 were unique, were analysed using an emulated HHL algorithm [16].…”

“…Section 3 describes the CFD test case. Section 4 briefly describes the emulated hybrid solver, for full details see [13]. Section 5 presents the main results and Section 6 draws conclusions.…”