Tensor network algorithms can efficiently simulate complex quantum many-body systems by utilizing knowledge of their structure and entanglement. These methodologies have been adapted recently for solving the Navier-Stokes equations, which describe a spectrum of fluid phenomena, from the aerodynamics of vehicles to weather patterns. Within this quantum-inspired paradigm, velocity is encoded as matrix product states (MPS), effectively harnessing the analogy between interscale correlations of fluid dynamics and entanglement in quantum many-body physics. This particular tensor structure is also called quantics tensor train (QTT). By utilizing NVIDIA's cuQuantum library to perform parallel tensor computations on GPUs, our adaptation speeds up simulations by up to 12.1 times. This allows us to study the algorithm in terms of its applicability, scalability, and performance. By simulating two qualitatively different but commonly encountered 2D flow problems at high Reynolds numbers up to 1 × 10 7 using a fourth-order time stepping scheme, we find that the algorithm has a potential advantage over direct numerical simulations in the turbulent regime as the requirements for grid resolution increase drastically. In addition, we derive the scaling χ = O ( poly ( 1 / ε ) ) for the maximum bond dimension χ of MPS representing turbulent flow fields, with an error ε , based on the spectral distribution of turbulent kinetic energy. Our findings motivate further exploration of related quantum algorithms and other tensor network methods.
Tensor network algorithms can efficiently simulate complex quantum many-body systems by utilizing knowledge of their structure and entanglement. These methodologies have been adapted recently for solving the Navier-Stokes equations, which describe a spectrum of fluid phenomena, from the aerodynamics of vehicles to weather patterns. Within this quantum-inspired paradigm, velocity is encoded as matrix product states (MPS), effectively harnessing the analogy between interscale correlations of fluid dynamics and entanglement in quantum many-body physics. This particular tensor structure is also called quantics tensor train (QTT). By utilizing NVIDIA's cuQuantum library to perform parallel tensor computations on GPUs, our adaptation speeds up simulations by up to 12.1 times. This allows us to study the algorithm in terms of its applicability, scalability, and performance. By simulating two qualitatively different but commonly encountered 2D flow problems at high Reynolds numbers up to 1 × 10 7 using a fourth-order time stepping scheme, we find that the algorithm has a potential advantage over direct numerical simulations in the turbulent regime as the requirements for grid resolution increase drastically. In addition, we derive the scaling χ = O ( poly ( 1 / ε ) ) for the maximum bond dimension χ of MPS representing turbulent flow fields, with an error ε , based on the spectral distribution of turbulent kinetic energy. Our findings motivate further exploration of related quantum algorithms and other tensor network methods.
Tensor networks (TNs), originally developed for simulating many-body quantum systems, provide a systematic framework for approximating high-dimensional fields. This is achieved by factorizing the field into interconnected tensors with small bond dimensions, thereby restricting the correlations captured across field bipartitions. Belonging to the family of TNs, the matrix product state (MPS) ansatz is utilized here as a reduced-order modeling framework to construct truncated representations of isotropic turbulent flow data. Two direct numerical simulation (DNS) datasets are considered: the hydrodynamic field of an incompressible three-dimensional flow, and a conserved Fickian scalar in a similar flow. Each field is encoded as an MPS through a sequence of singular value decompositions (SVDs) in which small singular values are discarded. The truncated representation is contracted back to the full grid, and the resulting reconstructed field is compared against DNS. An interleaved ordering of the spatial tensor indices of the transport variables is applied prior to decomposition in order to localize the dominant inter-tensor correlations. Velocity reconstructions achieve 99.8% fidelity using only 5% of the original DNS memory, while the scalar field reaches the same fidelity at 15% memory usage. A wide range of lower- and higher-order statistics, including velocity gradients, dissipation, and structure functions, are systematically examined. At these compression levels, the total kinetic energy and the scalar energy are both recovered within 0.2% relative error, while the mean dissipation and mean scalar dissipation remain within approximately 10% of the DNS generated values. These findings support the suitability of MPS for scalable reduced-order analysis of complex turbulent datasets and motivate further exploration of TN-based methods in computational turbulence.
Turbulent thermal convection governs heat transport in systems ranging from stellar interiors to industrial heat exchangers. Two-dimensional Rayleigh-Bénard convection serves as a paradigm for these flows, reproducing key features such as thin boundary layers, large-scale circulation, and sustained plume dynamics. While Matrix Product State (MPS) methods have demonstrated significant compression of isothermal turbulent fields, their application to buoyancy-driven flows with active thermal coupling has remained unexplored. We apply MPS to two-dimensional Rayleigh-Bénard convection with dynamical simulations up to Ra=1010. An a priori decomposition of DNS snapshots up to Ra=1011 shows that the bond dimension χ required to represent the flow fields grows without saturation, in contrast to the plateauing of χ reported for velocity fields in isothermal 2D turbulence. Crucially, however, dynamical simulations solving the governing equations directly in the compressed MPS format at fixed χ show that the χ required to recover statistical observables, such as the Nusselt number, scales significantly more favorably with Ra than the a priori complexity suggests. At Ra=1010, a relative error of 1.8% in the mean Nusselt number is achieved with a nearly 9-fold reduction in degrees of freedom, using a χ comparable to that required at Ra=109. Spectral analysis confirms the progressive recovery of spatial and temporal scales with increasing χ. These findings establish MPS as a scalable tool for simulating thermally driven turbulence, suggesting the method may remain viable for investigations of the ultimate regime at substantially higher Ra.
Quantum-inspired algorithms are an emerging class of algorithms for computational fluid dynamics (CFD) with potentially favourable scaling for large problems compared to classical methods. However, their applications have been limited to incompressible flows due to arithmetic limitations, which are addressed in this work. This work introduces the first true end-to-end quantum-inspired computational fluid dynamics (QICFD) solver for direct numerical simulation of the compressible Navier-Stokes equations, that is, all arithmetic operations, as well as initialisation and post-processing, are undertaken in the tensor train (TT) format. Importantly, new division and square-root algorithms using TTs enable the use of Sutherland's law for viscosity, and leverages recent improvements in TT arithmetic algorithms for reduced runtime and memory cost. The new QICFD solver is validated by comparison with the classical CFD solver HiPSTAR and by way of a challenging fluid-flow test case, the low resolution Taylor-Green Vortex (TGV) at Mach numbers of 0.8 and 0.1. The TGV test case is a transient turbulent case that is very sensitive to accumulating errors, yet our QICFD solver achieves excellent agreement with the classical CFD reference. This work demonstrates the correctness of the new TT division and square-root algorithms, and that QICFD is capable of compressible flow simulations. The new QICFD solver is also able to perform simultaneous simulations, running multiple TGV-like cases initialised differently in parallel with reduced extra cost. Finally, the demonstrated TGV test case reveals additional challenges of QICFD as well as highlight the need for future advances to make TT methods viable for industrially-relevant conditions.
This paper presents a novel approach to simulating complex fluid dynamics, specifically turbulent flows, utilizing quantum-inspired algorithms. Traditional numerical methods, such as Navier-Stokes solvers, often struggle with the computational demands of accurately representing turbulence, leading to significant processing times and limitations in practical applications. We introduce a simulation strategy based on quantum amplitude estimation (QAE), a technique inspired by quantum computing principles, to accelerate the solution process. The core idea is to approximate the probability amplitude associated with different fluid states, thereby reducing the number of iterations required to reach a stable solution. We demonstrate, through theoretical analysis and a simplified numerical framework, that QAE offers a promising path toward more efficient and accurate simulations of turbulent flows, potentially bridging the gap between classical computational fluid dynamics (CFD) and the computational benefits anticipated from full-scale quantum simulation. This work lays the groundwork for future research exploring the full potential of quantum-inspired algorithms in tackling complex fluid dynamics problems.
Direct numerical simulation (DNS) of turbulent reactive flows has been the subject of significant research interest for several decades. Accurate prediction of the effects of turbulence on the rate of reactant conversion, and the subsequent influence of chemistry on hydrodynamics remain a challenge in combustion modeling. The key issue in DNS is to account for the wide range of temporal and spatial physical scales that are caused by complex interactions of turbulence and chemistry. In this work, a new computational methodology is developed that is shown to provide a viable alternative to DNS. The framework is the matrix product state (MPS), a form of tensor network (TN) as used in computational many body physics. The MPS is a well-established ansatz for efficiently representing many types of quantum states in condensed matter systems, allowing for an exponential compression of the required memory compared to exact diagonalization methods. Due to the success of MPS in quantum physics, the ansatz has been adapted to problems outside its historical domain, notably computational fluid dynamics. Here, the MPS is used for computational simulation of a shear flow under non-reacting and nonpremixed chemically reacting conditions. Reductions of 30% in memory are demonstrated for all transport variables, accompanied by excellent agreements with DNS. The anastaz accurately captures all pertinent flow physics such as reduced mixing due to exothermicity & compressibility, and the formation of eddy shocklets at high Mach numbers. A priori analysis of DNS data at higher Reynolds numbers shows compressions as large as 99.99% for some of the transport variables. This level of compression is encouraging and promotes the use of MPS for simulations of complex turbulent combustion systems.
Resolving unsteady transport phenomena in geometrically complex domains is traditionally constrained by polynomial scaling of computational cost with spatial resolution. While methods based on tensor-network data representations or matrix-product states (MPS) data encodings have emerged as a technique to systematically reduce degrees of freedom, existing formulations do not extend to complex geometries and complex flow physics. Both capabilities are offered by lattice Boltzmann methods, for which we develop a generalized MPS formulation. This development marks a paradigm shift from classical methods that rely on explicit grid refinement for data reduction. Instead, our approach exploits non-local correlations in the MPS representation to systemically compress the global fluid state directly without modifying the underlying grid. We benchmark the proposed solver against classical LBM using three-dimensional flows through structured media and vascular geometries. The results confirm that the MPS formulation reproduces the reference solution with high fidelity while achieving compression ratios exceeding two orders of magnitude, positioning tensor networks or MPS encodings as a scalable paradigm for continuum mechanics on high-performance GPU hardware.