Quantum Algorithms for Nonlinear Differential Equations - QuANDiE

From designing Boeing 787 Dreamliners and studying the behaviour of space plasmas to creating the beautiful underwater world in "Finding Nemo", computational fluid dynamics (CFD) is instrumental to understanding the movement of liquids and gases (collectively, fluids). CFD encompasses a range of computer simulations for solving the equations of motion that often cannot be calculated by hand. The solutions allow us to find the fluid density, velocity, pressure, temperature, and chemical concentrations in relation to time and space. Many industries rely on CFD tools due to their reliability, versatility and affordability. However, one major challenge is the turnaround time. In CFD, we select specific points in the fluid volume at which we calculate its properties (velocity, pressure, etc) at different time steps. The more points we use, the longer the turnaround time needed to complete the calculations. This is most apparent when we study large systems or need a highly accurate description of the fluid. Although it is common practice to run on supercomputers (such as the Met Office's Cray XC40, used for studying weather and climate), these and other conventional computers are reaching physical limits to their computational power. In this context, quantum computers are being developed which promise to be much more powerful for some types of computations.

In this project, we will adapt two types of fluid simulation algorithms to run on quantum computers, and determine whether this will allow us to beat the classical computational limitations. We will take apart two specialised CFD methods: lattice Boltzmann (LB); and smoothed particle hydrodynamics (SPH); then examine how we can replace each step of the calculation with quantum-based procedures. Each component, from how to represent the fluid in a quantum computer to imposing boundary conditions that define the problem to measuring to extract the result, requires different techniques in a quantum setting. This dissection will also allow us to propose hybrid solvers that combine quantum-based techniques at the centre with supercomputers to scale up to larger sized simulations.

Equally important, we must determine the solution accuracy. If acceptable, does the quantum component provide any appreciable advantage? The crucial final step of our work will be to determine whether our newly developed quantum algorithms can be made useful as quantum computers become available. Furthermore, LB and SPH can be used to solve other types of equations that arise in scientific and industrial problems. The unique properties of quantum mechanics mean that our proposals potentially offer major advantages that can benefit engineering industries in addition to scientific discovery, weather forecasts and computer animations. Decreased computational times means lower costs, smaller carbon footprint and another strong contribution towards the UK's National Quantum Technologies Programme.