Benchmarking Quantum and Classical Algorithms for the 1D Burgers Equation: QTN, HSE, and PINN

TL;DR

The paper benchmarks three emerging solver families—Quantum Tensor Networks (QTN), Hydrodynamic Schrödinger Equation (HSE), and Physics-Informed Neural Networks (PINN)—against classical baselines to solve the 1D Burgers' equation, a canonical nonlinear PDE with shock formation. It finds QTN delivering high-precision solutions ($L_2 \sim 10^{-7}$) with near-constant runtime by leveraging entanglement compression, while HSE is efficient on demand but hampered by readout and dense-spectral costs, and PINN remains mesh-free yet limited by spectral bias to moderate accuracy ($L_2 \sim 10^{-1}$). The results reveal a regime-dependent landscape: quantum-inspired or native approaches can offer representational advantages at low entanglement but do not outperform classical solvers in the tested non-fault-tolerant setting, due to entanglement growth, readout bottlenecks, and optimization challenges. The work highlights the current maturity boundaries for quantum fluid dynamics and points to hardware deployment, higher-dimensional extension, and cost profiling as essential future steps toward practical quantum advantage.

Abstract

We present a comparative benchmark of Quantum Tensor Networks (QTN), the Hydrodynamic Schrödinger Equation (HSE), and Physics-Informed Neural Networks (PINN) for simulating the 1D Burgers' equation. Evaluating these emerging paradigms against classical GMRES and Spectral baselines, we analyse solution accuracy, runtime scaling, and resource overhead across grid resolutions ranging from $N=4$ to $N=128$. Our results reveal a distinct performance hierarchy. The QTN solver achieves superior precision ($L_2 \sim 10^{-7}$) with remarkable near-constant runtime scaling, effectively leveraging entanglement compression to capture shock fronts. In contrast, while the Finite-Difference HSE implementation remains robust, the Spectral HSE method suffers catastrophic numerical instability at high resolutions, diverging significantly at $N=128$. PINNs demonstrate flexibility as mesh-free solvers but stall at lower accuracy tiers ($L_2 \sim 10^{-1}$), limited by spectral bias compared to grid-based methods. Ultimately, while quantum methods offer novel representational advantages for low-resolution fluid dynamics, this study confirms they currently yield no computational advantage over classical solvers without fault tolerance or significant algorithmic breakthroughs in handling non-linear feedback.

Figures (12)

• Figure 1: Graphical representations of (a) a vector, (b) a fourth-order tensor, (c) matrix multiplication, (d) the expression in Equation (2), (e) the singular value decomposition (SVD) of a tensor, and (f) the definition of an isometry. Adapted from the original source, fig. 1.
• Figure 2: Caption
• Figure 3: architecture: Schematic architecture of the HSE implementation and time evolution of the state.
• Figure 4: Figure shows the relative $L_2$ error for grid sizes from $N=4$ to $N=128$
• Figure 5: HSE L2 Error convergence with Trotter step size $dt$.