###### Expert contributor for this post

Dr. Ljubomir Budinski

Quantum Algorithm Researcher

##### Advancements in quantum computing are paving the way for a new age of simulations.

In this article, we will briefly address the limitations of today’s simulations and highlight - mainly from a computational fluid dynamics (CFD) perspective - how quantum computing combined with the appropriate algorithms can help us reach new heights in these simulations, eventually moving R&D more within the digital scope.

#### The major problem in classical computation

Modern simulations deal with many moving parts mapped to so-called computational points. These computational points can represent, for example, a flowing liquid or moving air.

Each of these computational points is assigned with at least four variables to be solved for: three velocity components and, for example, temperature.

This, in and of itself, is computationally exhaustive for even modern computers when solving for a larger domain, say, the flow of a river. To keep the computation time reasonable, fewer computational points are used, hindering the accuracy of the result and the level of details in the final solution.

Another problem arises when we want to add additional components to the model; tracking sediments in the rivers or pollution in the air, for example. These new components yet again place a heavy burden on the process by adding a whole new set of independent variables to solve for.

If we want to account for all the variables and achieve a precise solution using a sufficient number of computational points while keeping the computation time reasonable, the current methods just don’t cut it; new systems are called for - regarding both software and hardware.

#### Quantum computing offers the solution — eventually

Quantum devices operate in quantum states. These quantum states are exponential by nature, meaning, n number of qubits can encode 2n states.

As you can imagine, this exponential nature quickly increases the number of possible quantum states unfathomably large. For reference, with around 256 qubits, we could assign a quantum state for each atom in the observable universe.

Now, the question that has been keeping researchers busy for years: how to take advantage of this exponentially large, so-called Hilbert space?

One inherent problem in simulating with quantum devices is the issue of linearity. Quantum devices work on the principles of quantum mechanics, which are linear by nature. The quantum state can be thought of as a vector in the Hilbert space, with elements representing, for example, pressure, velocity, etc.

What’s essential is that quantum mechanics only allows for linear transformations of the vectors.

The problem, however, is that the equations that need to be solved for the simulations (in fluid dynamics, these would be the Navier-Stokes equations) are nonlinear by nature. Combining the linear nature of quantum mechanics with the non-linearity of the CFD equations is something yet to be fully resolved. This, however, is a topic that is being researched, with developments being made each day.

#### NISQ era

Quantum computers rely on keeping the qubits stable inside the quantum chip. Even the slightest noise or dissipation can result in decoherence and, ultimately, errors in the computations performed.

Quantum devices with qubits stable enough to carry out the sufficient computations needed to achieve true quantum advantage don’t, unfortunately, yet exist. However, before running the algorithms that solve non-linear equations on fault-tolerant quantum machines, we can achieve speed-ups in simulations by, for example, solving linear equations using so-called NISQ devices.

NISQ (Noisy intermediate-scale quantum) refers to the current - transformational - era of quantum computers, where the qubits in the quantum chip are not quite advanced enough to establish a reliable source for the desired computational power of a fully fault-tolerant quantum machine.

Working IQM Quantum Computer installed in Espoo, Finland. (Source: IQM, licensed under CC BY-SA 4.0)

What can be achieved, however, is considerable speed-ups in solving linear equations with so-called variational quantum algorithms (VQA).

These VQA:s are a hybrid solution that can solve some parts of the problem using quantum technology while using classical computers to optimize the algorithm. They are already proven to provide speed-ups in some research papers; however, using VQA:s in real-world applications, for example, in the finite element method (FEM) - is still a matter of research.

In the current NISQ era, VQA:s are something worth pursuing, and the Quanscient team is making progress transferring them into real-world use.

#### In conclusion

By harnessing the immense power of quantum devices, we can simulate exact models of even large domains in a fraction of the time compared to any supercomputer today. When applied to, for example, electromagnetism or CFD, these simulations can replicate reality with impeccable precision, helping us transform much of R&D to the digital scope, saving time and resources. This could revolutionize design processes, enabling speed-ups in the product development phase, leading to much faster product output.

While advancements in the field are made as we speak, running the suitable algorithms through fault-tolerant quantum devices to achieve the aforementioned results is still ahead of us.

Even though quantum computing is still in its infancy, the profound effects it will have on simulations as we know them make them, at the very least, something to keep an eye on.