⚛️ quantum physics

Reconstructing fluid velocity fields from sparse sensors using a variational quantum algorithm

This paper proposes a variational quantum algorithm that reconstructs fluid velocity fields governed by nonlinear PDEs from sparse measurements by encoding the entire spacetime solution into a single quantum state and jointly optimizing a cost function that balances data fidelity with physics-informed constraints.

Original authors: Nhat-Quang Nguyen, Mohammad Mehedi Hasan Akash, Kourosh Shoele, Yanzhu Chen, Huixuan Wu

Published 2026-09-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Nhat-Quang Nguyen, Mohammad Mehedi Hasan Akash, Kourosh Shoele, Yanzhu Chen, Huixuan Wu

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

Imagine trying to understand the flow of a river by looking at only a handful of floating leaves. You can see where those specific leaves are at any given moment, but the water between them, the swirls forming in the gaps, and the currents moving upstream remain a mystery. This is the daily reality for scientists studying fluid dynamics, from weather patterns to blood flow in arteries. They often have access to only a few sensors scattered across a vast area, yet they need to know the speed and direction of the fluid at every single point in space and time. Reconstructing the full picture from such sparse clues is a notoriously difficult puzzle because the rules governing how fluids move are complex and change rapidly.

For decades, researchers have relied on powerful classical computers to fill in these gaps, using mathematical models that simulate the physics of the flow. However, as the need for higher resolution grows, these simulations become incredibly expensive and slow. A new approach has emerged from the intersection of physics and quantum computing, offering a different way to tackle this problem. Instead of trying to calculate the fluid's behavior step-by-step, moving forward in time like a movie playing frame by frame, this new method attempts to solve for the entire history of the flow at once. It treats the fluid's velocity across space and time as a single, unified object. By encoding this massive amount of information into a quantum system, researchers can use the unique properties of quantum mechanics to find the solution more efficiently.

In a recent study, a team of researchers from Florida State University proposed a new way to reconstruct these fluid velocity fields using a variational quantum algorithm. Their goal was to take the limited data from a few sensors and, combined with the known laws of physics, rebuild the complete map of how the fluid moves. They focused on two specific mathematical models that describe fluid behavior: the Burgers equation, which captures how waves steepen and break, and the Kuramoto–Sivashinsky equation, which describes more chaotic and turbulent patterns. These equations are famous for being difficult to solve because they involve strong non-linear interactions, meaning small changes can lead to wildly different outcomes.

The researchers developed a strategy that encodes the entire solution—every point in space and every moment in time—into a single quantum state. Think of this state as a complex, multi-dimensional container holding the answer. Instead of updating this container one second at a time, the algorithm adjusts the settings of the quantum computer all at once to find the configuration that best fits both the sensor data and the physical laws. The computer is guided by a cost function, a kind of scorecard that penalizes the solution if it disagrees with the sensor readings or if it violates the physics equations. The algorithm iteratively tweaks the quantum settings to lower this score, eventually converging on a solution that satisfies both constraints simultaneously.

To test their idea, the team ran numerical simulations on a classical computer that mimicked how a real quantum device would behave. They set up scenarios where sensors were placed at only a small fraction of the possible locations—sometimes as few as two or three points out of dozens—covering no more than 25 percent of the spatial grid. Despite this extreme lack of data, the algorithm successfully reconstructed the full velocity field. In the simulations, the reconstructed flow matched the known "true" flow with a high degree of accuracy, often achieving errors as small as one hundredth of the flow's magnitude. This level of precision was reached even when the fluid was moving through a low-viscosity regime, where the flow becomes sharp and turbulent, a condition that usually makes reconstruction very difficult.

The study compared two different ways of representing the fluid on the quantum computer. The first approach, called the real-space method, mapped the fluid directly to specific points on a grid, much like a digital image maps pixels. The second approach, known as the Galerkin-reduced-basis method, represented the fluid as a combination of smooth, wave-like patterns. The researchers found that for the smoother flows they tested, the wave-like representation was particularly effective, often producing more accurate results with fewer computational resources. This suggests that choosing the right way to describe the fluid's structure is just as important as the quantum algorithm itself.

One of the most significant aspects of this work is how it handles time. Traditional methods often struggle when the starting conditions of a fluid are unknown or when data is only available for a specific window in the middle of a process. The new method does not require a known starting point. Because it solves for the entire timeline simultaneously, it can work backward and forward from the sparse sensor data to fill in the gaps. In their simulations, the researchers demonstrated this by reconstructing flows where the sensors only recorded data during a short, arbitrary period, yet the algorithm still recovered the full history of the motion.

The results suggest that this spacetime encoding scheme offers a compact and powerful framework for understanding complex fluid dynamics. While the work is currently limited to simulations and one-dimensional models, it points toward a future where quantum devices could help scientists visualize invisible flows in real-world experiments. The researchers noted that the success of the method depends heavily on the design of the quantum circuit used to represent the solution. They used a standard, hardware-friendly design for their tests, but they suspect that tailoring the circuit to the specific physics of the problem could lead to even better performance.

Ultimately, this study provides a proof of concept that quantum algorithms can be used to solve inverse problems in fluid mechanics, where the goal is to deduce the whole from the part. By combining the constraints of physical laws with sparse measurements, the algorithm can recover a globally consistent picture of a fluid's motion. The researchers emphasize that while noise in real quantum hardware and the complexity of optimization remain challenges, the approach opens a new door for extracting meaningful information from limited data. It offers a promising path forward for fields ranging from meteorology to medical imaging, where seeing the full picture from a few scattered clues is essential.

Problem Statement
Reconstructing high-dimensional fluid velocity fields from sparse sensor measurements is a challenging inverse problem. The governing equations are typically nonlinear partial differential equations (PDEs), and observations are often limited to a small fraction of the spatial domain over a time window. Traditional approaches, such as Physics-Informed Neural Networks (PINNs) or time-marching variational quantum algorithms (VQAs), face limitations. PINNs can require prohibitively large networks for multiscale features, while standard time-marching VQAs are resource-intensive and error-prone, particularly when an initial condition is unknown or unavailable. The specific challenge addressed here is reconstructing a globally consistent spacetime solution for nonlinear PDEs (specifically fluid velocity fields) using only sparse, time-resolved measurements without a known initial state.

Methodology
The authors propose a Variational Quantum Algorithm (VQA) that encodes the full discrete spacetime solution into a single variational quantum state, rather than marching sequentially in time. The method operates by minimizing a composite cost function that balances two constraints:

Physics Constraint ( $C_{PDE}$ ): A term penalizing violations of the governing PDE (specifically the Burgers and Kuramoto–Sivashinsky equations) across the spacetime grid.
Measurement Constraint ( $C_{meas}$ ): A term penalizing the mismatch between the reconstructed field and sparse sensor data collected at fixed spatial locations over the entire time window.

The solution is represented using two distinct spatial encoding strategies:

Real-Space Approach: The quantum state amplitudes correspond directly to field values at discrete spatial grid points. Spatial derivatives are approximated using finite-difference operators.
Galerkin-Reduced-Basis Approach: The field is expanded in a truncated basis (specifically a sine Fourier basis for periodic domains). The quantum state encodes the time-dependent expansion coefficients. This approach allows for analytical evaluation of spatial derivatives within the basis, avoiding finite-difference errors and potentially offering spectral convergence for smooth flows.

The variational ansatz is a hardware-efficient circuit composed of single-qubit rotations ( $R_y$ ) and CNOT gates arranged in a brickwall pattern. To evaluate the nonlinear terms in the cost function (e.g., products of amplitudes), the authors utilize a Quantum Nonlinear Processing Unit (QNPU) and cyclic index-shifting operators. The cost function is measured on quantum hardware, and the parameters are optimized classically using the L-BFGS-B optimizer.

Key Contributions

Spacetime Encoding: The paper introduces a VQA framework where the entire spacetime solution is encoded in a single state. This reduces the circuit preparation overhead for measurement residuals from $O(N_t)$ (in time-marching schemes) to $O(1)$ , where $N_t$ is the number of time steps.
Handling Unknown Initial Conditions: The method is designed to work without a known initial condition, relying instead on the global consistency enforced by the PDE residual and sparse sensor data across the time window.
Dual Representation Framework: The authors implement and compare both real-space and Galerkin-reduced-basis encodings, providing specific quantum circuit constructions for evaluating cost function terms in both regimes, including the handling of convolution terms in the Galerkin approach via auxiliary states and the Quantum Bluestein's Algorithm (QBA) for discrete sine transforms.
Algorithmic Implementation: The paper details the specific quantum circuits required to evaluate PDE violations and measurement mismatches for both the Burgers and Kuramoto–Sivashinsky equations, demonstrating how to handle nonlinear terms and boundary conditions within a variational setting.

Results
The method was validated through numerical simulations on one-dimensional Burgers and Kuramoto–Sivashinsky (KS) equations under moderate ( $\nu=0.05$ ) and low ( $\nu=0.008$ ) viscosity regimes.

Accuracy: The reconstructed velocity fields achieved root-mean-square errors (RMSE) on the order of $10^{-2}$ to $10^{-3}$ using sensor data from no more than 25% of the spatial grid points.
Comparison of Approaches: In the moderate-viscosity regime with sinusoidal initial profiles, the Galerkin-reduced-basis approach generally achieved higher accuracy (e.g., RMSE $\approx 8.9 \times 10^{-3}$ vs. $2.67 \times 10^{-2}$ for the real-space approach in the KS case) with fewer parameters, likely due to the compact representation of smooth solutions in the chosen basis.
Robustness: The method successfully reconstructed fields in low-viscosity regimes with steep gradients and in scenarios where the measurement window began at an arbitrary time without access to the initial condition.
Scalability: The number of distinct measurement circuits required for cost function evaluation was found to be independent of the total number of qubits, scaling only with the order of the finite-difference stencils or basis truncation.

Significance and Claims
The authors claim that their spacetime-encoded VQA offers a compact framework for reconstructing nonlinear PDE dynamics from sparse observations. By avoiding explicit time-marching and encoding the full spacetime domain in a single state, the method addresses the resource constraints of near-term quantum devices while handling the specific challenge of missing initial conditions. The paper suggests that this approach enables the recovery of globally consistent spacetime fields from spatially sparse observations, extending the capabilities of traditional diagnostic methods that typically yield only partial information. The authors note that while the current results use a problem-agnostic ansatz, future work could improve performance by incorporating problem-specific structures into the circuit architecture. They also acknowledge that direct extraction of the full solution state is inefficient, and future utility lies in extracting specific physical observables from the reconstructed fields.

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