Putting quantum computing to the test

From forecasting how smoke disperses through a city to predicting heat transfer inside a turbine, engineers turn to a workhorse mathematical model known as the advection-diffusion equation. The equation describes how a quantity such as temperature or concentration is carried by a flow (advection) while also spreading through diffusion. It is a foundation for modeling in fluid mechanics, heat and mass transfer, combustion, and many other transport problems.

Advection-diffusion equations, however, might require immense computing power that can strain even the most powerful classical computers. Running simulations, especially in fine detail and repeatedly, can be prohibitively time consuming and costly. Yet these simulations can dramatically improve how engineers design everything from airplanes to energy systems.

Researchers at the University of Pittsburgh Swanson School of Engineering and Pitt’s School of Computing and Information have teamed up with scientists from Ames National Laboratory / Iowa State University, Boeing Research & Technology, and the Naval Nuclear Laboratory to take a new approach. They have tested whether powerful quantum computers, which process information differently than classical systems, can solve these equations.

Led by the Swanson School’s Juan Jose Mendoza Arenas, Peyman Givi, and Hirad Alipanah, the researchers developed and evaluated three algorithms, demonstrating the potential of quantum computers to solve real-world engineering problems.

The research, which shed important new light on the emerging field of quantum computing, is detailed in the paper, “Quantum dynamics simulation of the advection-diffusion equation,” published on December 19, 2025, in Physical Review Research (DOI: 10.1103/ndc3-bdwt).

Testing a new kind of computing

“Classical computers operate with a binary logic of ones and zeros, which limits their ability to simulate complex systems,” said Mendoza Arenas, assistant professor in the Department of Mechanical Engineering and Materials Science. “Quantum computers function under the laws of quantum physics and have the potential to run complex equations more quickly, using less computational power. The challenge is to reformulate classical equations to run on these newer quantum systems.”

Indeed, to run advection-diffusion equations on a quantum computer, the team had to translate a physical process into something the new computational language could understand—what is known as a Hamiltonian. The Hamiltonian serves as an engine governing how the system evolves quantum states.

The intensive work involved breaking physical space into small points and encoding the value at each point to a quantum state. Then, the team developed new algorithms that quantum systems could process.

To test the potential of simulating a one-dimensional model using a quantum computer, the researchers formulated and assessed three approaches:

1. Trotterization, a strategy that accurately approximates the mathematical time evolution dictated by the Hamiltonian. This approach, while providing the most accurate results, was the most resource intensive and impractical on current quantum hardware.
2. Variational Quantum Time Evolution (VarQTE), a hybrid quantum-classical computing approach that proved more practical than Trotterization but less precise.
3. Adaptive Variational Quantum Dynamics Simulation (AVQDS), an extension of the VarQTE strategy that starts simple and adds components as needed. This approach was most adaptable and was the only one used to simulate a two-dimensional flow.

The researchers ran each approach on quantum simulators and real systems and then compared their results to direct numerical simulation (DNS), a high-accuracy classical benchmark.

“We found that in an idealized, noise-free simulation, the quantum methods reproduced the same solution as the gold-standard classical simulation,” said Alipanah, a PhD student in Computational Modeling and Simulation and the first author of the paper.

“Through our research, we have developed incredibly promising algorithms,” added Givi, Distinguished Professor in the Department of Mechanical Engineering and Materials Science. “We’ve demonstrated the potential of quantum computing to solve some of the most complex, vexing problems in engineering.”