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Quantum-Inspired Computational Fluid Dynamics for Transient Turbulent Compressible Flows

This paper introduces and validates the first complete quantum-inspired computational fluid dynamics solver capable of simulating transient turbulent compressible flows using tensor train formats, demonstrating its accuracy against classical methods and its efficiency in running parallel simulations while highlighting remaining challenges for industrial applications.

Original authors: Shang Xian Matthew Lee, Melissa Kozul, Muhammad Usman, Martin Sevior, Matthew L. Sims-Goh, Richard D. Sandberg

Published 2026-08-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shang Xian Matthew Lee, Melissa Kozul, Muhammad Usman, Martin Sevior, Matthew L. Sims-Goh, Richard D. Sandberg

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Fluid dynamics is the science of how liquids and gases move, a field that underpins everything from the design of jet engines to the prediction of weather patterns. To understand these flows, scientists rely on powerful computer simulations that solve complex equations describing how matter behaves. For decades, the most accurate way to model these flows has been to break the fluid down into a massive grid of tiny points and calculate the changes at each point over time. However, as the flows become more turbulent and the speeds approach the speed of sound, the number of points required grows so large that even the world's fastest supercomputers struggle to finish the job in a reasonable time. This limitation forces engineers to use simplified models that can miss critical details, leading to designs that are overly cautious or less efficient than they could be.

A new approach is emerging that borrows ideas from the physics of quantum systems to tackle this problem. Instead of treating the fluid as a giant, unwieldy list of numbers, this method represents the data as a connected chain of smaller, manageable pieces. This structure, known as a tensor train, allows the computer to compress the information, much like a highly efficient file format, so that it can handle complex patterns without needing to store every single detail explicitly. While this "quantum-inspired" technique had shown promise for simulating slow-moving, incompressible fluids, it hit a wall when faced with fast-moving, compressible flows where the density of the fluid changes. The mathematical operations required to handle these changes, specifically division and calculating square roots, were previously impossible to perform efficiently within this compressed format, leaving a major gap in the method's capabilities.

In a recent study, researchers have successfully bridged this gap by developing new mathematical tools that allow division and square-root calculations to be performed directly on these compressed data chains. By creating these specific algorithms, the team built the first complete solver capable of simulating compressible, turbulent flows using this quantum-inspired approach. They tested their new system against a state-of-the-art classical computer program using a challenging test case known as the Taylor-Green vortex. This test involves a swirling, turbulent flow that is notoriously difficult to simulate because tiny errors can quickly grow and ruin the results. The researchers ran simulations at two different speeds, one where the flow was moving at 80% of the speed of sound and another at 10% of the speed of sound. The results showed that their new solver matched the classical reference program with remarkable precision, keeping errors below one percent for the majority of the simulation time. This agreement confirmed that their new division and square-root algorithms work correctly and that the entire system can now handle the complex physics of compressible fluids.

Beyond simply matching existing results, the study demonstrated a unique advantage of this new method: the ability to run multiple simulations at the same time with very little extra cost. In traditional computing, running two separate simulations usually requires twice the computing power and time. However, because the quantum-inspired method encodes multiple scenarios into a single, larger data structure, the researchers were able to run two different versions of the flow simulation simultaneously. When they compared the time it took to run these two cases together against running just one, the time increased by only about 20%, whereas a classical computer would have taken twice as long. This suggests that as the number of simulations needed grows, this method could become increasingly efficient compared to traditional approaches, offering a potential pathway to solve problems that are currently too expensive to tackle.

Despite these successes, the researchers also identified a significant hurdle that must be overcome before this technology can be used for real-world industrial problems. The efficiency of the method depends on a property called the "rank," which essentially measures how complex the patterns in the fluid are. For the test cases they used, the rank remained low enough to be manageable, but when they analyzed more complex, high-speed flows with higher turbulence, the rank grew rapidly. In these more difficult scenarios, the amount of memory required to store the data could become too large for current computers to handle. The study suggests that while the method is highly effective for certain types of flows, its broader application depends on finding ways to keep this complexity low or on developing even more efficient ways to manage the data.

The work represents a significant step forward in the field of computational fluid dynamics, proving that quantum-inspired algorithms can now handle the full range of equations needed for compressible flows. By solving the specific mathematical bottlenecks of division and square roots, the researchers have opened the door to simulating high-speed aerodynamics and acoustic phenomena with a new kind of efficiency. While challenges remain regarding how the method scales with extreme turbulence, the ability to run multiple simulations simultaneously with minimal overhead offers a compelling glimpse into a future where complex fluid problems might be solved faster and more accurately than ever before. The next steps will involve refining these tools to handle the messy, high-complexity conditions found in actual aircraft engines and industrial machinery, testing whether the theoretical advantages can hold up in the most demanding environments.

Technical Summary: Quantum-Inspired Computational Fluid Dynamics for Transient Turbulent Compressible Flows

Problem Statement
Computational Fluid Dynamics (CFD) faces a fundamental scalability barrier when simulating high-fidelity, transient turbulent flows, particularly for compressible regimes. While Direct Numerical Simulation (DNS) offers the highest accuracy by resolving all spatial and temporal scales without modeling, its computational cost scales prohibitively with the Reynolds number ( $Re$ ), following a power law of approximately $Re^{2.05}$ for grid points. Current industrial practices often rely on Reynolds-Averaged Navier–Stokes (RANS) approaches, which sacrifice accuracy for efficiency, sometimes resulting in significant prediction errors (e.g., up to 400 K for jet engine blade temperatures).

Quantum-inspired algorithms, specifically those utilizing Tensor Train (TT) networks, offer a potential pathway to overcome these limitations by representing high-dimensional data with exponentially fewer parameters, provided the data exhibits low entanglement or is "well-behaved." However, previous applications of Quantum-Inspired CFD (QICFD) were restricted to incompressible flows. The transition to compressible flows requires handling variable density and temperature-dependent viscosity (via Sutherland's law), which necessitates element-wise division and square-root operations. Existing QICFD methods lacked robust, efficient algorithms for these specific arithmetic operations within the TT format, often forcing the contraction of the full state vector (defeating the scaling advantage) or relying on inefficient optimization-based division methods.

Methodology
The authors developed the first complete QICFD solver capable of Direct Numerical Simulation (DNS) for the compressible Navier–Stokes equations, where all arithmetic operations are performed directly on tensors in the TT format.

Tensor Train Encoding: The solver encodes multivariate functions (density, velocity, energy) as Tensor Train States (TTS) using "quantics" representation. This maps a discretized grid of size $2^N$ to a chain of $N$ tensors, where correlations are managed via virtual indices (bond dimensions, $\chi$ ).
Novel Arithmetic Algorithms: The core methodological innovation involves the development of robust division and square-root algorithms for TTs:
- Division: Implemented via the Newton–Raphson method to compute the reciprocal of the divisor, followed by element-wise multiplication. This reduces division to scalar multiplication, addition, and multiplication, all of which are well-established in TT arithmetic. An initial polynomial approximation is used to seed the iteration.
- Square-Root: Also implemented using the Newton–Raphson method, leveraging the new division algorithm to solve $x^2 - D = 0$ .
- These algorithms enable the calculation of Sutherland's law for viscosity and the retrieval of primitive variables (e.g., velocity) from conservative variables (e.g., momentum/density).
Solver Architecture: The QICFD solver mimics the classical finite-difference solver HiPSTAR. It employs a conservative formulation of the Navier–Stokes equations with skew-symmetric splitting of derivatives to enhance numerical stability. Time integration is performed using a fourth-order Runge–Kutta (RK4) scheme, and spatial derivatives use fourth-order central difference stencils.
Simultaneous Simulation: The authors introduced a method to encode multiple distinct flow fields (e.g., different initial conditions) into a single TTS by adding an extra coordinate. This allows for the parallel execution of multiple simulations within a single tensor network structure.

Key Contributions

First Compressible QICFD Solver: The work presents the first complete QICFD solver for the compressible Navier–Stokes equations, operating entirely in the TT format without contracting to a dense state.
New Arithmetic Routines: The introduction of accurate and efficient element-wise division and square-root algorithms for TTs, which are critical for handling variable density and temperature-dependent viscosity in compressible flows.
Validation of Compressibility: Demonstration that QICFD can successfully simulate transient turbulent compressible flows with excellent agreement to classical solvers.
Simultaneous Simulation Capability: A novel demonstration of running multiple distinct simulations concurrently within a single TT framework, showing marginal cost increases compared to classical parallelization.

Results
The solver was validated against the classical CFD solver HiPSTAR using low-resolution Taylor–Green Vortex (TGV) test cases at Reynolds number $Re=800$ and Mach numbers $Ma=0.8$ and $Ma=0.1$ .

Accuracy: The QICFD solver achieved excellent agreement with the classical reference. For the $Ma=0.8$ case, the error in total kinetic energy was below 0.1% and enstrophy error was below 0.6% for the majority of the simulation time ( $t < 15$ ). Errors increased slightly toward the end of the simulation (up to ~2.1% for $Ma=0.8$ and ~4.6% for $Ma=0.1$ ), attributed to the chaotic nature of turbulence amplifying small numerical errors.
Simultaneous Simulations: When running two TGV cases with different initial conditions simultaneously, the QICFD solver incurred only a 20% increase in runtime (140s to 170s), whereas a classical solver would require a 200% increase (running two separate simulations).
Rank Dependence ( $\chi$ ): Analysis of the bond dimension $\chi$ revealed that it scales as a power law with the Reynolds number. For high-fidelity cases ( $Re=1600$ ), the required $\chi$ (up to 2700 for strict accuracy) currently exceeds the practical limits of the solver due to memory constraints ( $O(N\chi^3)$ memory scaling). However, lower $\chi$ settings (e.g., $\chi \approx 500$ ) are manageable with current hardware, though they introduce truncation errors.

Significance and Claims
The paper claims to demonstrate the correctness of the new TT division and square-root algorithms and establishes that QICFD is capable of performing compressible flow simulations. The work represents a significant advancement in the field, expanding the applicability of quantum-inspired methods beyond incompressible flows to aerospace and acoustic applications where density variations are critical.

The authors highlight the potential for QICFD to address the engineering need for massively parallel simulations (e.g., parametric sweeps or design optimization) with favorable scaling. However, the paper remains modest regarding immediate industrial viability. It acknowledges that for highly turbulent, high- $Re$ flows, the required bond dimension $\chi$ may be too large for current hardware. The authors suggest that future work must investigate the $\chi$ characteristics of specific industrial problems, potentially leveraging steady-state statistics or lower-fidelity models (RANS/LES) where lower $\chi$ might suffice, and further optimize the solver for memory and performance.

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