Magic of the Well: assessing quantum resources of fluid dynamics data

TL;DR

This work tackles the quantum-resource costs of representing fluid-dynamics data for CFD using matrix product states (MPS). By quantifying entanglement with $\tilde{S}_{vN}$ and non-stabilizerness with the stabilizer Rényi entropy $\tilde{m}_2$ (at $\alpha=2$) for a 2D periodic shear-flow dataset, the authors map how resource requirements evolve across Reynolds $R$, Schmidt $S$, and initial width $w$, and how mesh resolution and data sign structure influence representational cost. A key finding is that the shear width $w$ can separate regimes of resource-efficient versus resource-intensive evolution, with the two quantum resources tracking each other over time, suggesting comparable computational demands under their respective resource theories. The results provide a diagnostic framework for choosing quantum-inspired CFD approaches and highlight practical preprocessing steps—such as data positivity and appropriate encoding—to reduce resource costs and guide hybrid quantum-classical solvers. This work lays groundwork for scalable quantum-inspired fluid dynamics and informs when stabilizer-TN solvers may outperform classical methods.

Abstract

We investigate the quantum resource requirements of a dataset generated from simulations of two-dimensional, periodic, incompressible shear flow, aimed at training machine learning models. By measuring entanglement and non-stabilizerness on MPS-encoded functions, we estimate the computational complexity encountered by a stabilizer or a tensor network solver applied to Computational Fluid Dynamics (CFD) simulations across different flow regimes. Our analysis reveals that, under specific initial conditions, the shear width identifies a transition between resource-efficient and resource-intensive regimes for non-trivial evolution. Furthermore, we find that the two resources qualitatively track each other in time, and that the mesh resolution along with the sign structure play a crucial role in determining the resource content of the encoded state. These findings offer useful guidelines for the development of scalable, quantum-inspired approaches to fluid dynamics.

Figures (13)

• Figure 1: Schematic representation of the tracer field $s_{ij}$ (top), which is then encoded as an MPS (bottom). The encoding of the shown example has a bond dimension $\chi=21$, which is $\approx 8\%$ of the possible maximum $\chi_{max}$. The tracer field is obtained as a solution of eq. \ref{['eq:pde_sf']}.
• Figure 2: Visual representation of the encoded data for the tracer field. From top to bottom: original data in ohana2024well; image obtained after a lossless MPS encoding (accuracy w.r.t. the original image, as measured through the root mean squared error, is $\approx 99.7 \%$); image obtained after high compression of the MPS encoding (fine details are lost).
• Figure 3: Resource phase diagrams of the shear flow data illustrating how entangled (bottom row) or magic (top row) content each simulation has. Each column represents a different Reynolds number with the top row showing maximum normalized entanglement entropy $\tilde{S}_{vN}$ and the bottom row showing maximum normalized magic content $\tilde{m}_2$. Each plot shows the Schmidt number $S$ as a function of initial shear width factor $w$. The shear width factor is inversely proportional to how sharply the transition between the two flows are. Interpolation was used between data points.
• Figure 4: Comparison of the normalized non-stabilizerness $\tilde{m}_2(t)$ and normalized entanglement entropy $\tilde{S}_{vN}(t)$ for the tracer field as a function of time for different values of $R$, $S$, and initial shear width $w \in [1,3,4]$. The insets highlight the initial and final stages of the evolution, while $\chi$ indicates the maximum bond dimension attained in each case for the MPS encoding. The global truncation error for $t=128$ of each simulation is shown in Fig. \ref{['fig:all_trunc_err']} in the appendix.
• Figure 5: Top panels: scaling of the normalized non-stabilizerness and entanglement entropy with the number of grid points at early, intermediate and late evolution times. Bottom panels: RMSE in the interpolation from the original grid size to the rescaled ones. All plots refer to the solutions of eq. \ref{['eq:pde_sf']} for the tracer field.