Author:Jukka Knuutinen

Expert contributor:Dr. Ljubomir Budinski

Expert contributor:Dr. Ljubomir Budinski

SIMULATIONS
-️ March 28, 2022

Quantum computing can enable us to reach unseen heights in the field of computer-created simulations. The immense computational power of a quantum device - when harnessed fully - allows us to create precise models of even large domains.

The accuracy of these models would also be so precise that the current design processes of companies working in electromagnetics, mechanics, or fluid dynamics, could be entirely reshaped by allowing them to skip multiple phases of prototype testing.

Unfortunately, none of this comes without some significant challenges embedded in the physics of quantum mechanics itself.

This article will go over these inherent issues while also shedding light on the possible solutions currently being rigorously researched.

Quantum devices work on the principles of quantum mechanics, which are linear by nature (besides the measurements). Technically speaking, the time evolution of the quantum states is linear, meaning that the dynamics of the quantum space is subjected to the space-time transformation of the quantum states, which is linear.

All of this is to say, that quantum mechanics only allows for linear transformations in the vector space.

However, accurate physics simulations tend to often require solving non-linear equations, and herein lies the problem.

So the challenge now is solving the non-linear equations with a system that is, at least for the most parts, inherently linear.

The solution that most likely first comes to mind is to simply linearize the equations. Sure enough, this is a compelling approach currently under research.

A technique called the Carleman linearization is used precisely for this process.

For example, in the Navier-Stokes equations, you could apply the Carleman linearization producing linear differential equations. From there, you can use the finite difference method to create a system of algebraic equations that can then be solved by using some of the quantum algorithms for solving the linear system of equations (HHL or VQLS.)

When the Carleman linearization process was first successfully tested with tailor-made problems for a quantum device, it was effectively announced as a sensation, leading to a false impression that quantum computers can now solve non-linear equations.

This, unfortunately, may never be true due to the underlying quantum mechanics.

It should be mentioned that the usage of the Carleman linearization regarding the Navier-Stokes equations is still a huge question mark with minimal research available. Furthermore, applying the Carleman linearization to complex non-linear equations can lead to inaccurate results, as much of the information is lost in the process itself.

Before its application to quantum devices, the Carleman linearization’s functionality with the Navier-Stokes equations should first be researched using classical methods.

One proposed method to overcome the linearization problem is the lattice Boltzmann method (LBM). This is a method for solving computational fluid dynamics (CFD) problems based on a mesoscopic approach.

In LBM - much like in the finite element method (FEM) - you dissect the model into computational points that you then solve for. However, the primary advantage in LBM is that, unlike in FEM, LBM allows for the computational points to be solved independently of each other.

This so-called parallelization makes the computations more efficient, allowing for massive speed-ups. The parallelization process makes LBM particularly appealing when dealing with quantum devices.

But, even though LBM is perhaps the most promising method, it still has some unresolved issues currently under research, namely, the same old issue of linearity.

LBM, in and of itself, has some non-linear terms that, as we know, cannot be solved with a quantum device. The number of these terms can, however, be reduced down to just one using the so-called stream vorticity formulation, but nonetheless, the problem of linearity remains.

There are also other formulations of LBM, with possibilities of eliminating the non-linear terms altogether, but as for now, this is purely hypothetical and a matter of research.

Also, a hybrid method is proposed, where the quantum device solves for the linear terms leaving the non-linear parts to a classical computer, but as with the others, more research is still needed to support the practicality of this approach.

The second major advantage of LBM is the fact that we can solve systems of non-linear equations without using a solver for linear system of equations. In the quantum setting, we simply implement the distribution function - in which LBM operates - in the quantum state, encode it here, manipulate the state, and get the solution.

The absence of linear solvers allows for more efficiency, speeding up the process.

- The linearity compatibility issue is a topic being researched, with multiple solutions proposed. Most of these rely on first linearizing the non-linear equations. This, however, is not optimal as we lose information during the process.
- One exciting approach also being researched here at Quanscient is the LBM, in which no linear solvers have to be used. Furthermore, the parallel nature of LBM allows for more efficient solution, resulting in speed-ups.
- LBM has lots of potential, although it has some issues currently under research, mainly regarding the issue of linearity
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