# Challenges in classical CFD and the promise of quantum computing

Computational Fluid Dynamics (CFD) is essential for understanding how fluids behave, but it faces challenges when tackling complicated problems like designing airfoils.

Accurately modeling airflow, particularly turbulence, demands significant computing power. Additionally, simulations involving multiple physical processes, like heat transfer alongside airflow, further strain computational resources.

Quantum computing's potential to speed up calculations could revolutionize how we approach CFD, especially in airfoil design.

In this article, we explore the challenges in traditional CFD and demonstrate how quantum computing offers a solution. We will discuss a quantum algorithm designed specifically for airfoil simulations and show its potential to efficiently tackle complex multiphysics problems.

By going beyond the current limitations of CFD, quantum computing could lead to more accurate and innovative airfoil designs, with potential benefits in other fields as well.

#### The challenge

Quantum computing aims to overcome these limitations by accelerating multiphysics simulations beyond the capabilities of classical methods.

In classical computing, expanding the complexity of the problem (finer mesh, multiphysics, complex flows) necessitates additional computational resources. There's a linear relationship between problem complexity and resource requirements.

Quantum computing seeks to break this linear dependency and achieve exponential speedup, providing a logarithmic relationship instead.

This advantage becomes particularly significant as the computational space grows.

#### The Airfoil Quantum Algorithm as a solution

The core idea is to develop a quantum algorithm capable of delivering exponential speedup for multiphysics simulations.

This requires identifying a numerical method suitable for solving differential equations on a quantum system.

Due to the inherent constraints of quantum mechanics, traditional numerical methods used in classical computing may not be efficient.

The solution lies in the lattice Boltzmann method (LBM), which aligns well with the quantum system due to its quantum-friendly steps: collision, propagation, and macros.

By leveraging LBM, we have created a quantum algorithm to address the airfoil + temperature problem.

The airfoil is simulated using a stream-vorticity approach, involving two lattice Boltzmann equations. Temperature is simulated with an additional equation, resulting in a total of three coupled equations.

##### Key innovations

This algorithm introduces several advancements:- It is the first instance of simulating a multiphysics problem on a quantum computer (simulator) by solving three coupled lattice Boltzmann equations.
- Certain parts of the algorithm, namely propagation and macros, are fully optimized, leading to exponential speedup through quantum properties like superposition and entanglement.
- The superposition principle enables the simulation of propagation and macros steps for all three LB equations in a single run.
- This quantum algorithm has been successfully applied to real-world scenarios with complex boundary conditions, demonstrating the feasibility of simulating complex multiphysics problems on quantum computers.

##### Leveraging quantum computers

The key lies in optimizing specific parts of the quantum lattice Boltzmann method to achieve logarithmic dependence and exponential speedup.

The propagation and macros steps have been particularly optimized, while the collision step still requires a hybrid approach due to its nonlinear nature.

The optimization is twofold:

- Logarithmic dependency is implemented between computational operations (quantum gates) and the number of states (data) those operations act upon.
- The superposition principle allows these optimal algorithms to be applied to all three lattice Boltzmann equations, enabling the execution of the propagation step for all three equations in a single run.

This concept can be extended further, simulating billions of propagation steps in one run, offering a novel approach to achieving quantum advantage. The same principle applies to the macros step.

##### Impact and applications

##### Improving airfoil design and performance

Quantum computers excel at solving problems intractable for classical computers.

As the complexity of these problems grows, the resources required for exact solutions on classical machines increase exponentially.

Quantum computers offer an efficient way to handle simulations and calculations in large computational spaces, delivering advantages in detail, computational power, and execution time.

The airfoil case serves as a demonstration of the quantum algorithm's potential, which extends beyond this specific application.

Broader applications

The discoveries made in this research go beyond airfoils and the automotive industry.

They can be applied to various sectors requiring multiphysics simulations. For example, a similar algorithm is being developed for acoustic quantum LBM.

The impact of this quantum algorithm is far-reaching and extends to any field necessitating multiphysics simulations.

Long-term implications

The long-term implications of this research are significant.

It opens doors to solving multiphysics problems that are currently beyond the reach of classical computers.

This capability has the potential to revolutionize aviation, transportation, and other fields by enabling simulations and computations in previously inaccessible computational spaces.