Exploiting the structure of turbulence with tensor networks

Turbulence is among the most important unsolved problems of physics. Its numerical solution is hindered by the extreme number of computational variables needed to accurately resolve the broad range of length-scales that become relevant during turbulent flow. It turns out that a similar challenge appears in a completely different branch of physics. Quantum many-body systems are described by elements of a vector space whose dimension grows exponentially with the number of particles, making direct simulation infeasible. However, the actual information contained in realistic quantum states is typically only polynomially large in the number of particles, which in principle makes it possible to represent them using a polynomial number of parameters. Tensor network methods do precisely this for quantum systems with local interactions by removing unrealised long-distance correlations from the solution space, which in turn enables accurate simulation of otherwise intractable quantum systems. In this thesis, a simple tensor network formalism known as the matrix product state (MPS) ansatz is transferred from quantum physics onto fluid dynamics and used to numerically examine two paradigmatic turbulent flows: the 2D temporally decaying jet and 3D collapse of the Taylor-Green vortex. We find both flows to be structured according to the classical, scale-local view of turbulence, where flow features of disparate scales are largely uncorrelated. We eliminate these unrealistic interscale correlations from the solution space through our MPS encoding of the velocity field, and then formulate a MPS algorithm for simulating turbulence. With this algorithm, we find that the incompressible Navier-Stokes equations can be accurately solved even when reducing the number of computational parameters by more than one order of magnitude compared to traditional direct numerical simulation. The outlook is threefold. Further work towards harnessing the power of tensor networks for turbulence simulation holds the promise of computational fluid dynamics calculations that are yet inconceivable in scale. Moreover, the close connection that our MPS algorithm has to quantum physics points towards the exciting prospect of solving the Navier-Stokes equations on a quantum computer. Finally, from a theoretical standpoint, this work also lays the foundations for studying the structures of turbulence using tensor network theory. One topic of particular interest here is what tensor network geometry is most appropriate for turbulence simulations and why. Answering this question will illuminate the structure of turbulence from a completely new angle, and perhaps help unravel the old riddle that is predicting the dynamics of turbulent flows

Parallel time-dependent variational principle algorithm for matrix product states

Combining the time-dependent variational principle (TDVP) algorithm with the parallelization scheme introduced by Stoudenmire and White for the density matrix renormalization group (DMRG), we present the first parallel matrix product state (MPS) algorithm capable of time evolving one-dimensional (1D) quantum lattice systems with long-range interactions. We benchmark the accuracy and performance of the algorithm by simulating quenches in the long-range Ising and XY models. We show that our code scales well up to 32 processes, with parallel efficiencies as high as 86%. Finally, we calculate the dynamical correlation function of a 201-site Heisenberg XXX spin chain with 1/r2 interactions, which is challenging to compute sequentially. These results pave the way for the application of tensor networks to increasingly complex many-body systems

A quantum-inspired approach to exploit turbulence structures

Understanding turbulence is key to our comprehension of many natural and technological flow processes. At the heart of this phenomenon lies its intricate multiscale nature, describing the coupling between different-sized eddies in space and time. Here we analyze the structure of turbulent flows by quantifying correlations between different length scales using methods inspired from quantum many-body physics. We present the results for interscale correlations of two paradigmatic flow examples, and use these insights along with tensor network theory to design a structure-resolving algorithm for simulating turbulent flows. With this algorithm, we find that the incompressible Navier–Stokes equations can be accurately solved even when reducing the number of parameters required to represent the velocity field by more than one order of magnitude compared to direct numerical simulation. Our quantum-inspired approach provides a pathway towards conducting computational fluid dynamics on quantum computers

A quantum-inspired approach to exploit turbulence structures

Understanding turbulence is key to our comprehension of many natural and technological flow processes. At the heart of this phenomenon lies its intricate multiscale nature, describing the coupling between different-sized eddies in space and time. Here we analyze the structure of turbulent flows by quantifying correlations between different length scales using methods inspired from quantum many-body physics. We present the results for interscale correlations of two paradigmatic flow examples, and use these insights along with tensor network theory to design a structure-resolving algorithm for simulating turbulent flows. With this algorithm, we find that the incompressible Navier–Stokes equations can be accurately solved even when reducing the number of parameters required to represent the velocity field by more than one order of magnitude compared to direct numerical simulation. Our quantum-inspired approach provides a pathway towards conducting computational fluid dynamics on quantum computers