Quantum Fluid Dynamics May Overcome Computing Limitations

Luca Cappelli and colleagues at Sapienza University of Rome in collaboration with Delft University of Technology and University of Trento, present a Schrödinger-Navier-Stokes (SNS) formulation, a quantum-like wave approach, to model classical dissipative fluids on quantum computers. Building on earlier, largely overlooked work, the group propose a strategy using the Hamilton-Jacobi formulation of fluid dynamics and a new tensor-network technique for Carleman embedding. Through classical emulation, they demonstrate the convergence and accuracy of their algorithm for Kolmogorov-like flows, representing a first step towards a quantum simulation of the full Navier-Stokes equations, including key elements like pressure gradients and viscosity.

Tensor networks enable laptop simulation of complex fluid dynamics with minimal memory

Computational memory requirements have fallen from approximately 1 gigabyte to 0.01 gigabytes using a new tensor-network technique. This dramatic decrease unlocks the possibility of simulating complex fluid dynamics on standard laptops, a feat previously unattainable due to the immense processing demands of traditional methods. Grounded in a wave-like formulation of the Navier-Stokes equations, the quantum algorithm accurately models fluid behaviour up to a dissipative time T, maintaining relative errors below 10−2 despite lacking systematic convergence. The Navier-Stokes equations, fundamental to describing fluid motion, traditionally require discretisation of both space and time, leading to a rapid increase in computational cost with higher resolution and longer simulation times. This is particularly problematic for turbulent flows, characterised by a wide range of spatial and temporal scales.

A crossover timescale, dependent on flow regime, dictates the accuracy limits of different Carleman truncation orders, allowing for optimised simulations balancing precision and computational cost. The Carleman embedding is a mathematical technique used to transform partial differential equations into a series representation, enabling efficient truncation and approximation. The choice of truncation order directly impacts the accuracy of the solution and the computational resources required. Simulations successfully ran on a standard laptop equipped with an M2 processor, a result previously impossible with the full Carleman matrix which would have demanded OGB of memory. Kolmogorov-like flows at Reynolds numbers of approximately 5.33 and 16 revealed a timescale influencing simulation accuracy, varying between T/4 and T/2 depending on the flow regime. Reynolds number is a dimensionless quantity that characterises the ratio of inertial forces to viscous forces within a fluid, with higher values indicating a greater tendency towards turbulence. The observed timescale limitation suggests that the algorithm’s accuracy is sensitive to the specific characteristics of the flow being simulated. With a time step of 0.01 for ν = 1/6, short-time simulations extending to T/4 showed that the fourth-order Carleman truncation reproduced reference solutions with errors below 10−3. This demonstrates a level of accuracy achievable within the limitations of the current implementation. Further investigation focused on the algorithm’s performance characteristics, specifically examining the impact of Carleman truncation order on both accuracy and computational expense, providing guidance for optimising simulations. The optimisation process involves finding the optimal balance between computational cost and desired accuracy, crucial for practical applications.

Quantum simulation offers potential for computationally efficient turbulence modelling

Accurate modelling of fluid dynamics remains vital for advances in areas like weather forecasting, climate modelling, aircraft design, and biomedical engineering, yet traditional computational methods demand ever-increasing processing power. The demonstration of a quantum algorithm capable of simulating these flows offers a potential route to overcome these limitations. However, current results are confined to relatively simple, moderate-flow scenarios, and the algorithm’s ability to handle the chaotic complexity of turbulent flows remains unproven. The computational expense of simulating turbulence stems from the need to resolve all relevant scales of motion, a task that quickly becomes intractable for high Reynolds number flows. Furthermore, accurately capturing the intermittent and multiscale nature of turbulence presents significant challenges for any numerical method.

The approach builds on largely overlooked theoretical work from 1985, translating the governing Navier-Stokes equations into a quantum-like wave formulation. Specifically, Dietrich and Vautherin, in their 1985 paper published in the Journal de Physique, proposed a Schrödinger-like equation for fluid dynamics, laying the groundwork for the current research. This formulation leverages the mathematical similarities between the Schrödinger equation in quantum mechanics and the Navier-Stokes equations in classical fluid dynamics. A corresponding tensor-network technique was devised, sharply reducing the computational memory needed for simulations from gigabytes to megabytes. Tensor networks are a powerful tool for representing high-dimensional data in a compact and efficient manner, making them well-suited for simulating complex systems. Successful classical emulation of the algorithm for specific, simplified flows demonstrates its potential to overcome limitations imposed by traditional methods, particularly regarding the accurate representation of pressure and dissipation. The accurate representation of pressure gradients and viscous dissipation is crucial for capturing the essential physics of fluid flow. This technique offers a potential pathway to alleviate the computational pressures associated with modelling complex turbulence, though further work is needed to assess its scalability and applicability to more realistic scenarios. Future research will likely focus on extending the algorithm to handle more complex geometries, boundary conditions, and flow regimes, as well as exploring the potential benefits of implementing it on actual quantum hardware. The ultimate goal is to develop a computationally efficient and accurate method for simulating turbulent flows, enabling advances in a wide range of scientific and engineering disciplines.

Researchers have developed a new quantum algorithm based on a wave-like formulation of the Navier-Stokes equations, which describe classical fluid motion. This approach utilises a tensor-network technique to reduce the computational memory required for simulations, achieving savings from gigabytes to megabytes. The algorithm was successfully emulated on a classical computer using simplified flows at moderate Reynolds numbers, accurately representing pressure drops and velocity fields.