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arXiv:2604.00620

Quantum machine learning for the quantum lattice Boltzmann method: Trainability of variational quantum circuits for the nonlinear collision operator across multiple time steps

Antonio David Bastida Zamora, Ljubomir Budinski, Pierre Sagaut, Valtteri Lahtinen

quant-phphysics.flu-dynSubmitted Apr 1, 2026• Announced Apr 2, 2026• 0 citations
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Abstract

This study investigates the application of quantum machine learning (QML) to approximate the nonlinear component of the collision operator within the quantum lattice Boltzmann method (QLBM). To achieve this, we train a variational quantum circuit (VQC) to construct an operator $U$. When applied to the post-linear-collision quantum state $\ket{Ψ_i}$, this operator yields a final state $\ket{Ψ_f} = U\ket{Ψ_i}$ that successfully replicates the nonlinear collision dynamics derived from the Bhatnagar-Gross-Krook (BGK) approximation. Within this framework, we present two distinct architectures: the R1 model and the R2 model. The R1 model is designed for quantum simulations that involve multiple time steps without intermediate measurements, focusing on accurately capturing nonlinear dynamics in continuous evolution. In contrast, the R2 model is tailored to achieve the high-precision reconstruction of the nonlinear operator for a single time step with an unitary operator.

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This study investigates the application of quantum machine learning (QML) to approximate the nonlinear component of the collision operator within the quantum lattice Boltzmann method (QLBM). To achieve this, we train a variational quantum circuit (VQC) to construct an operator UUU. When applied to the post-linear-collision quantum state ∣Ψi⟩\ket{Ψ_i}∣Ψi​⟩, this operator yields a final state ∣Ψf⟩=U∣Ψi⟩\ket{Ψ_f} = U\ket{Ψ_i}∣Ψf​⟩=U∣Ψi​⟩ that successfully replicates the nonlinear collision dynamics derived from the Bhatnagar-Gross-Krook (BGK) approximation. Within this framework, we present two distinct architectures: the R1 model and the R2 model. The R1 model is designed for quantum simulations that involve multiple time steps without intermediate measurements, focusing on accurately capturing nonlinear dynamics in continuous evolution. In contrast, the R2 model is tailored to achieve the high-precision reconstruction of the nonlinear operator for a single time step with an unitary operator.
We attempt the use of a unitary operator to approximate the lattice Boltzmann collision operator. We use a modified amplitude encoding to bypass the renormalization that would have required classical processing at every step (thus eroding any quantum advantage to be had). We describe the hard-wiring of the lattice Boltzmann symmetries into the quantum circuit and show that, for the specific case of the cavity flow, approximating the nonlinear system is limited to low velocities. These findings may help us understand better the possibilities of nonlinear simulations on a quantum computer, and also pave the way for a discussion on how quantum machine learning might be harnessed to address more complex problems.
In the existing quantum algorithms for the lattice Boltzmann method, resolving the unitarity problem of the BGK collision operator from first principles remains a fundamental challenge, and a systematically constructed unitary alternative that reproduces the BGK relaxation structure to leading order has not yet been established in the literature. In this work, we propose a novel quantum lattice Boltzmann method (QLBM) based on the rotation collision operator, in which the non-unitary BGK relaxation is replaced by a unitary rotation in the amplitude space of the distribution function, and the corresponding gate-level quantum circuits are also constructed. Specifically, the rotation collision operator is developed for both the D2Q5 model of the convection-diffusion equation and the D2Q9 model of the incompressible Navier--Stokes equations. It is worth noting that, for the D2Q5 model, the equilibrium-state preparation circuit can be precompiled and reused, and the resulting unitary blocks can be sequentially composed once the state-dependent collision parameters are specified. Finally, some numerical experiments are performed to validate the developed QLBM, demonstrating its accuracy and effectiveness.
Quantum computing is rapidly advancing and offering new ways to improve artificial intelligence (AI).The most recent advancement is Quantum Machine Learning (QML), an exciting new field that involves the exploration of combining quantum computing principles with machine learning methods to solve complex problems that have high-dimensional problem-solving models that are impossible to address with classical models.The objective of this research was a descriptive-analytical examination of combining classical machine learning with quantum computing principles, specifically using Variational Quantum Classifiers (VQC) to train on a nonlinear XOR dataset, to assess the proposition of QML.The VQC was able to amongst quantum features, and learning how to classify using parameterized variational circuits to model nonlinear decision boundaries as models not achievable using classical linear classifiers.Training convergence was demonstrated with decreasing training and test losses, with the final model achieving test accuracy of nearly 0.85.The training curves and the decision boundary plots illustrated the model's ability to learn the more complex decision boundaries.This study demonstrated QML's ability to coexist with classical machine learning studies with the ability to exploit quantum entanglement and quantum parallelism.This paper concluded that QML has great potential to challenge the concerns of next generation applications in finance, healthcare, and secure communications and to optimize computational power and scalability.
Classical computational fluid dynamics (CFD) methods encounter significant computational bottlenecks in large systems and at high Reynolds number flows due to the rapid increase in degrees of freedom, even on flagship supercomputers. As quantum computers emerge, the Quantum Lattice Boltzmann Method (QLBM) has been proposed as a quantum-native alternative for solving fluid dynamics problems. However, until recently, nearly all studies have relied on classical quantum simulators rather than real quantum hardware. This study presents a benchmark of a one-dimensional QLBM implementation executed on three different IBM Quantum hardware. We design and test a quantum circuit for the advection-diffusion equation using amplitude encoding, a Bhatnagar-Gross-Krook (BGK)-type collision operator, and a streaming step with periodic boundary conditions. Our numerical experiments evaluate transpiled two-qubit gate count, total circuit depth, runtime, and accuracy metrics including maximum absolute error (L∞L_\inftyL∞​ norm) and root-mean-square error (RMSE). We find that the present collision and streaming operators dominate the overall cost and become the primary scalability bottleneck on current hardware. To mitigate the prohibitive cost of full quantum state tomography on hardware-based benchmarking exercises, we introduce a sampler-based measurement strategy that reconstructs macroscopic density fields directly from computational-basis measurements. This approach substantially reduces measurement overhead while preserving the expected advection-diffusion dynamics on hardware. These results provide a quantitative assessment of QLBM performance in the noisy intermediate-scale quantum (NISQ) regime and identify the principal algorithmic and experimental limitations that must be addressed for QLBM to become viable for fluid dynamics.
This study presents a Quantum Machine Learning (QML) architecture for perfectly classifying the Iris flower dataset. The research addresses improving classification accuracy using quantum models in machine-learning tasks. The objective is to demonstrate the effectiveness of QML approaches, specifically the Variational Quantum Circuit (VQC), Quantum Neural Network (QNN), and Quantum Support Vector Machine (QSVM), in achieving high performance on the Iris dataset. The proposed methods result in perfect classification, with all models attaining accuracy, precision, recall, and an F1-score of 1.00. The main finding is that the QML architecture successfully achieves flawless classification, contributing significantly to the field. These results underscore the potential of QML in solving complex classification problems and highlight its promise for future applications across various domains. The study concludes that QML techniques can offer transformative solutions in machine learning tasks, particularly those leveraging VQC, QNN, and QSVM.
A Boltzmann machine is a powerful tool for modeling probability distributions that govern the training data. A thermal equilibrium state is typically used for the Boltzmann machine learning to obtain a suitable probability distribution. The Boltzmann machine learning consists of calculating the gradient of the loss function given in terms of the thermal average, which is the most time-consuming procedure. Here, we propose a method to implement the Boltzmann machine learning by using noisy intermediate-scale quantum devices. We prepare an initial pure state that contains all possible computational basis states with the same amplitude, and we apply a variational imaginary time simulation. Readout of the state after the evolution in the computational basis approximates the probability distribution of the thermal equilibrium state that is used for the Boltzmann machine learning. We perform the numerical simulations of our scheme and confirm that the Boltzmann machine learning works well. Our scheme leads to a significant step toward an efficient machine learning using quantum hardware.