Quantum Technique Solves Complex Equations in Consistent Time, Unlike Rivals

Researchers are increasingly exploring novel computational methods to tackle complex problems in fluid dynamics, and this study benchmarks the performance of quantum and machine learning techniques against established classical algorithms for solving the one-dimensional Burgers’ equation. Vanshaj Kerni from the Department of Physics at the Indian Institute of Technology Roorkee, Abdelrahman E. Ahmed from Alexandria University, and Syed Ali Asghar from the Department of Physics at the University of Karachi et al. compare Tensor Networks (QTN), the Hydrodynamic Schrödinger Equation (HSE), and Physics-Informed Neural Networks (PINN) with classical GMRES and Spectral methods. This research is significant because it rigorously assesses whether these emerging paradigms can offer a computational advantage in simulating fluid flow, revealing a clear performance hierarchy and highlighting the challenges in achieving speed-ups without substantial algorithmic improvements.

Results demonstrate that the QTN solver surpasses conventional methods in capturing intricate fluid behaviours, particularly those involving shock fronts, by effectively leveraging entanglement compression.

The core achievement lies in the exceptional precision attained by the QTN solver, reaching an L2 error of 10 -7 . Furthermore, the QTN solver exhibits remarkable near-constant runtime scaling, suggesting efficient handling of increasing computational demands.
This contrasts sharply with the Spectral HSE method, which suffered from catastrophic numerical instability at higher resolutions, diverging significantly at a grid resolution of 128. The research meticulously compares the performance of these algorithms, focusing on metrics crucial for assessing their potential in the quantum computing landscape.

Evaluations spanned grid resolutions ranging from 4 to 128, providing a comprehensive understanding of each method’s strengths and weaknesses. While PINNs offer flexibility as mesh-free solvers, their performance was constrained by spectral bias, hindering their ability to achieve high accuracy. Ultimately, this study confirms that, despite offering novel representational advantages, current quantum-inspired methods do not yet provide a computational advantage over classical solvers without significant algorithmic advancements or the implementation of fault tolerance.

However, the demonstrated precision of 10 -7 achieved by the QTN solver establishes a new benchmark and highlights the potential of tensor network-based approaches for tackling complex fluid dynamics problems. This breakthrough paves the way for future investigations into enhancing these algorithms and exploring their applicability to more intricate, high-dimensional simulations.

Comparative performance of quantum and classical solvers for one-dimensional Burgers’ equation simulations reveals potential advantages and disadvantages of each approach

A quantum tensor network (QTN) solver achieved a level of precision significantly exceeding other tested methods when simulating the one-dimensional Burgers’ equation. The QTN solver attained superior precision, registering an L2 error of 10 -7 , demonstrating remarkable near-constant runtime scaling and effective utilisation of entanglement compression to accurately capture shock fronts.

The study employed a discrete representation of the 1D Burgers’ equation, utilising finite differences to approximate spatial derivatives. Algorithms were implemented to solve the equation for various grid resolutions, systematically increasing the number of grid points from 4 to 128 to assess scalability.

Classical GMRES and spectral methods served as baselines for comparison, providing established benchmarks for accuracy and computational cost. The finite-difference HSE implementation proved robust throughout the tested resolutions, while the spectral HSE method exhibited catastrophic numerical instability at higher resolutions, diverging significantly at N = 128.

Physics-informed neural networks were implemented as mesh-free solvers, offering flexibility in handling complex geometries. However, these networks stalled at a lower accuracy tier, achieving an L2 error of approximately 10 -1 , limited by spectral bias when compared to grid-based methods. The QTN solver distinguished itself through its ability to maintain high precision even as grid resolution increased.

This was achieved by leveraging quantum-inspired analytical tensor factorisations to compress the solution space, effectively reducing the computational burden associated with high-resolution simulations. The research meticulously tracked runtime scaling and resource overhead for each method, providing a comprehensive assessment of their performance characteristics.

Researchers implemented the QTN solver using a classical computing framework, exploiting tensor network algorithms to represent and manipulate the solution space. The core innovation lay in the application of entanglement compression techniques, which allowed the solver to efficiently capture the complex shock fronts that develop in the Burgers’ equation.

This involved systematically reducing the number of parameters needed to represent the solution, thereby minimising computational cost and memory requirements. The study carefully monitored entanglement growth during the simulation, assessing the effectiveness of the compression strategy. The performance of the QTN solver was directly linked to its ability to maintain a manageable level of entanglement while preserving solution accuracy.

This contrasts with the spectral HSE method, which suffered from instability due to uncontrolled growth of high-frequency components. The experimental setup involved solving the 1D Burgers’ equation with a Riemann step initial condition, a standard benchmark for evaluating numerical solvers. The kinematic viscosity, ν, was varied to control the Reynolds number, influencing the strength of the shock formation.

The accuracy of each solver was quantified using the L2 error norm, measuring the difference between the computed solution and a high-resolution reference solution. Runtime was measured using wall-clock time, providing a direct comparison of computational efficiency. Resource overhead was assessed by tracking memory usage and the number of floating-point operations performed. This detailed analysis revealed that the QTN solver not only achieved superior accuracy but also exhibited near-constant runtime scaling, indicating its potential for handling even higher-resolution simulations.

Quantum tensor networks outperform classical and machine learning approaches for Burgers’ equation simulation in certain regimes

Achieving a level of precision at 10 -7 , a Quantum Tensor Network (QTN) solver demonstrates markedly superior performance in simulating the one-dimensional Burgers’ equation. The QTN solver’s precision of 10 -7 was achieved alongside near-constant runtime scaling, indicating efficient handling of shock fronts through entanglement compression.

In contrast, a Finite-Difference HSE implementation proved robust, while the Spectral HSE method exhibited catastrophic numerical instability at higher resolutions, diverging significantly at N = 128. PINNs, functioning as mesh-free solvers, were limited to lower accuracy tiers, reaching only approximately 10 -1 , due to spectral bias when compared to grid-based approaches.

This work benchmarked the performance of these solvers using the 1D viscous Burgers’ equation, initialised with a Riemann step profile to induce shock formation. Error quantification was performed using the relative L2 norm, providing a precise measure of solution accuracy. The time integration horizon for all grid-based methods was consistent, based on the Courant-Friedrichs-Lewy condition with a CCFL value of 0.1.

The research explicitly distinguishes between quantum-inspired (QTN) and quantum-native (HSE) approaches, focusing on metrics relevant to the quantum computing community, including circuit depth, entanglement scaling, and noise sensitivity. This level of accuracy, combined with near-constant runtime scaling, highlights QTN’s potential for efficiently capturing complex flow features like shock fronts through entanglement compression.

However, the study also identifies an “entanglement barrier” where resource demands increase as sharper discontinuities require finer resolution, limiting quantum advantage in highly turbulent flows. The HSE method, while conceptually promising, is constrained by the computational cost of updating the non-linear potential, and spectral implementations proved unstable at higher resolutions.

PINNs offer flexibility as mesh-free solvers but currently lack the precision of grid-based methods. The authors acknowledge that current quantum-inspired and quantum-native methods do not yet offer a computational advantage over classical solvers for fluid dynamics, and future work will focus on deploying these algorithms on physical quantum hardware and extending them to higher dimensions to address more complex fluid phenomena.