Quantum Algorithms for CFD simulations
Simulation and modelling, enabled by high performance supercomputers, have transformed the way products are designed and engineered. Central to this progress has been Computational Fluid Dynamics (CFD).
To simulate one full revolution of a stateoftheart unsteady CFD simulation of a gas turbine compressor using almost 5 billion cells takes 5.5hrs on 65,536 cores of the UK’s national supercomputer ARCHER2. The world’s current fastest supercomputer, Frontier, has a total of 8,730,112 cores and has achieved 1.102 Exaflop/s. By contrast, the compressor simulation requires only a 2 Petaflop/s system. So the question is?
Will quantum computers beat classical supercomputers?
QEC Project is committed to belief that only faulttolerant (FT) quantum computers will be able to outperform classical ones at the scales described above. Hence, much of Technical Challenge 3 is focused on emulation using the ATOS QLM at the STFC Hartree Centre to understand how Quantum Linear Equation Solvers (QLES) are likely to perform in a hybrid classicalquantum environment. RollsRoyce’s ambition is to have proven algorithms and capabilities ready to deploy as soon as viable FT devices are available.
Hybrid quantumclassical CFD
methodology with benchmark HHL solution

Evaluate quantum linear equation solvers.

Progress on the use of Quantum Singular Value Transformation (QSVT).

Matrix preparation, i.e., decomposing the CFD matrix into an efficient circuit implementation.

State preparation, i.e., loading the righthandside vector.

Measurement, i.e., returning the solution vector, x, to the outer nonlinear solver.

Optimising the measurements and observables to provide higher fidelity measurements of the solution is an important topic.

The goal is to find log(N) circuit implementations that deliver output solutions that retain the convergence rate of the outer nonlinear iterations.

Having optimised the circuits in emulation mode we hope to demonstrate the smaller test matrices on a physical device.

Perform hypothetical scaling studies and cost models to create a roadmap to quantum advantage, informed by the research in Technical challenges 1 and 2.
As part of the project collaboration, a set of test matrices have been produced for a specific type of CFD solver which uses a pressure correction equation to solve the mass conservation equation. The salient features of the matrices are listed in the Table 1 below. RollsRoyce have published an initial analysis of these matrices using the Harrow, Hassidim, Lloyd (HHL) QLES.
Table 1. Dimensions, sparsity, eigenvalue range and condition number for pressure correction equations. The HHL column gives an estimation of the number of logical qubits needed by HHL
Hybrid CFD Solver processes used in the experiment
Comparison of classical and HHL solutions for the 9×9 mesh matrix after 10 iterations