A Quantum Inspired Approach to Exploit Turbulence Structures

Nikita Gourianov; Michael Lubasch; Sergey Dolgov; Quincy Y. van den Berg; Hessam Babaee; Peyman Givi; Martin Kiffner; Dieter Jaksch

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A Quantum Inspired Approach to Exploit Turbulence Structures

Nikita Gourianov, Michael Lubasch, Sergey Dolgov, Quincy Y. van den Berg, Hessam Babaee, Peyman Givi, Martin Kiffner, Dieter Jaksch

arXiv:2106.05782v3Jun 10, 2021physics.flu-dynquant-ph

TL;DR

The paper addresses how to quantify and exploit multiscale correlations in turbulent flows. It uses quantum-inspired interscale analysis and tensor-network representations to simulate turbulence accurately with reduced computational space, while noting that MPS efficiency can degrade when distant-scale correlations matter.

Problem
Turbulence involves complex coupling between different length scales, motivating methods that quantify and exploit interscale correlations.
Method
The paper combines quantum-inspired interscale-correlation analysis with matrix product state tensor networks to represent and simulate turbulent velocity fields.
Results
The MPS algorithm captures jet large-scale dynamics comparably to DNS at compression ratio 1 : 49, while corresponding URDNS results clearly deviate from DNS.
Takeaways & Limitations
The structure-resolving tensor-network approach can reduce computational requirements for turbulence simulation and points toward Navier-Stokes implementations on quantum computers.
Takeaways & Limitations
MPS can become numerically inefficient when correlations between distant length scales are relevant, requiring a very large bond dimension χ for accuracy.

Abstract

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Understanding turbulence is the key to our comprehension of many natural and technological flow processes. At the heart of this phenomenon lies its intricate multi-scale nature, describing the coupling between different-sized eddies in space and time. Here we introduce a new paradigm for analyzing the structure of turbulent flows by quantifying correlations between different length scales using methods inspired from quantum many-body physics. We present 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 within a computational space reduced by over an order of magnitude compared to direct numerical simulation. Our quantum-inspired approach provides a pathway towards conducting computational fluid dynamics on quantum computers.

RESULTS

The paper measures interscale correlations in turbulent flows and uses their scale-local structure to construct compressed tensor-network simulations. Across 2-D and 3-D cases, MPS representations preserve flow dynamics at substantially lower computational space than DNS, while their efficiency depends on correlation scaling and network suitability.

Quantifying interscale correlations: Schmidt numbers quantify interscale correlations by counting retained terms needed to represent velocity fields across coarse–fine grid bipartitions.The decomposition retains orthonormal spatial structures weighted by descending Schmidt coefficients.
Quantifying interscale correlations: χ99 = 25 represents the 2-D jet DNS solutions with 99% L2 accuracy, and the retained correlations remain well below their maximal values.The corresponding d99 values are entirely contained within the compressed region shown in Fig. 1b.
Quantifying interscale correlations: χ99 = 207 remains below the 3-D maximal-correlation bound, while 2-D χ99 saturates for Re ≳200 and 3-D χ99 follows a power law.These trends indicate bounded interscale correlations in the studied 2-D case but increasing correlations in 3-D.
Tensor network algorithm: The MPS algorithm encodes each velocity component as products of scale-associated matrices, with bond dimension χ controlling compression.Keeping χ constant as grid resolution increases makes the number of MPS parameters scale logarithmically with total grid points without removing length scales.
2-D temporally developing jet: MPS captures the 2-D jet dynamics at compression ratios of 1 : 64, 1 : 16, and 1 : 8, while χ = 74 and 118 are practically indistinguishable from DNS.At equal numerical variables, MPS remains accurate longer than URDNS, whose lower-resolution simulations fail earlier.
3-D Taylor–Green vortex: For the 3-D Taylor–Green vortex, MPS remains comparable to DNS at compression 1 : 49, whereas corresponding URDNS results visibly deviate even at compression 1 : 25.MPS with χ = 128 and 192 also dissipates energy more accurately than URDNS, especially for t/T0 ≥1.4.
Computational scaling: MPS complexity scales as ∼χ^4 log M, compared with DNS scaling ∼Re3K/4 log Re, and outperforms DNS when γ < 3K/16 for χ ∼Re^γ.The studied 2-D case suggests γ ≈0 and exponential speedup at sufficiently large Re, whereas the TGV has γ ≈0.71.
Scope and extensions: MPS can become inefficient when distant-scale correlations matter, motivating alternative tensor networks such as TTNs or MERA with different connectivity trade-offs.The paper also identifies compressible flows, transported scalars, and high-dimensional reactive-flow PDFs as future application areas.

Supplementary Information: A Quantum Inspired Approach to Exploit

The supplementary information accompanies the paper with author, affiliation, and preprint metadata.

The paper lists eight authors, including Nikita Gourianov and Dieter Jaksch.
The authors are affiliated with institutions in the United Kingdom, United States, Singapore, and Germany.
The work is identified as arXiv:2106.05782v3, dated 4 Jul 2022.

1. ADDITIONAL RESULTS

The additional-results section examines interscale correlations in DNS data for the TDJ and TGV flows using Schmidt spectra and von Neumann entanglement entropy.

The section expands the main-text results by analyzing Schmidt spectra and von Neumann entanglement entropy for DNS solutions of the TDJ and TGV flows.

A. Schmidt spectra

The supplementary Schmidt-spectrum analysis compares normalized spectra across times and grid bipartitions for the TDJ and TGV flows, while marking the truncations used for 99% accuracy.

The supplementary spectra include d99(n, t) contours to assess whether truncation discards relevant interscale correlations.
The TDJ spectrum uses DNS data at Re=1000 across times t/T0 = 0.25, 0.75, 1.25, 1.75 and 9 bipartitions of a 1024 × 1024 grid.
The Schmidt coefficients are sorted in descending order and normalized so that their squares sum to 1.
The black dashed lines mark d99(n, t), the truncation values used in the main text.
The TGV spectrum uses DNS data at Re=800 across times t/T0 = 0.2, 0.8, 1.4, 2 and 7 bipartitions of a 256 × 256 × 256 grid.

B. Entanglement entropy

The supplementary material interprets entanglement entropy across length-scale bipartitions and details how matrix product states encode discretized flow fields through scale-organized tensors.

B. Entanglement entropy: Von Neumann entanglement entropy is computed from the Schmidt spectrum λ_α(n, t).
B. Entanglement entropy: In the TDJ, entropy shifts toward lower n over time, consistent with an inverse energy cascade toward coarser length scales.
B. Entanglement entropy: In the TGV, entropy increases at larger n over time, consistent with a direct cascade toward finer length scales.
B. Entanglement entropy: The TGV entropy increases with time across all bipartitions, accompanying disorder from collapse into worm-like vortical structures.
MPS encoding: The grid contains M = 2^(KN) points, and binary index bits are grouped by length scale to organize the flow representation.
MPS encoding: An MPS approximates each discretized velocity component using matrices A^ω_n whose bond dimension χ controls the captured interscale correlations.
MPS encoding: At maximal χ, the MPS becomes exact and uses Q = 2^(KN) = M parameters after accounting for gauge degrees of freedom.
Quantum analogy: The same MPS formalism represents quantum wavefunctions with polynomially many variables when χ is limited, without significant accuracy loss for many area-law systems.

B. Mathematical solution for initial δ function

The Burgers’ equation solution evolves from a δ-function initial condition, with limiting behavior that becomes triangular as ν→0 and Gaussian as ν grows large. These limits provide the mathematical basis for analyzing MPS representations.

B. Mathematical solution for initial δ function: The section analyzes the hump solution generated by a δ-function initial condition for Burgers’ equation.The derivation uses asymptotic expansions of the complementary error function and split integrations.
B. Mathematical solution for initial δ function: As ν→0, the solution approaches a triangular-wave form.This limit is obtained by analyzing the solution separately across x≤x0 and x>x0.
B. Mathematical solution for initial δ function: As ν grows large, the solution approaches a Gaussian, which becomes uniform with vanishing amplitude as ν→∞.The Gaussian limit follows from the large-ν asymptotics of the derived solution.

C. Matrix product state representation of general solution

The general solution is represented using matrix product states, whose convergence and bond dimension depend on the solution’s structure. The construction extends from exact low-bond-dimension representations to the variational MPS algorithm used for flow fields.

C. Matrix product state representation of general solution: The discretized δ function has an exact MPS representation with bond dimension χ=1.Grid points are indexed by binary digits, and the peak position is encoded through the MPS factors.
C. Matrix product state representation of general solution: For finite ν, the Burgers’ solution is holomorphic and therefore admits an exponentially convergent MPS approximation.The ν→0 triangular wave is exactly representable with χ=3, while the large-ν Gaussian has an exponentially convergent approximation.
C. Matrix product state representation of general solution: The MPS description requires exponentially fewer variables than a standard finite-difference scheme for the considered initial-value problem.This conclusion follows from exponential convergence in the number of MPS variables.
C. Matrix product state representation of general solution: The triangular-wave solution has a non-redundant MPS representation with bond dimension 3.The construction uses Heaviside and unit-vector MPS factors and reduces redundant tensor columns during the recursion.
C. Matrix product state representation of general solution: The flow solver minimizes a Navier–Stokes cost function variationally within the MPS manifold using local tensor updates and global sweeps.Each local problem is solved through linear equations, conjugate-gradient descent, and canonical-centre shifts analogous to DMRG.

D. Theoretical computational scaling

The MPS algorithm’s cost is governed primarily by bond dimension and grows only logarithmically with the number of grid points. Its quartic bond-dimension dependence is the main computational bottleneck, with a possible lower-cost approximation noted.

D. Theoretical computational scaling: The canonical-centre shift costs O(χ^3), while exact Hadamard products make each local minimization scale as χ^4+m2χ^3.The quartic term dominates the local minimization cost at large χ.
D. Theoretical computational scaling: The supplementary scaling experiment measures CPU time for 10 time-steps across five χ values and eight grid sizes.The simulations use the 2-D Navier–Stokes equations and report CPU time in seconds.
D. Theoretical computational scaling: The algorithm scales as O(χ^4 log M) per time-step.This follows from global sweeps, local minimizations, canonical-centre shifts, and M=2^KN.
D. Theoretical computational scaling: The quartic dependence on χ is identified as a central bottleneck of the algorithm.The authors note that replacing exact Hadamard products with cross approximation could reduce the relevant cost from O(χ^4) to O(χ^3), if accuracy loss is not severe.

E. Demonstration of computational scaling

The demonstration measures the MPS algorithm’s CPU time for 10 time-steps of the 2-D Navier–Stokes equations across different grid sizes and bond dimensions.

E. Demonstration of computational scaling: CPU time is plotted for 10 simulated time-steps across five χ values and eight grid sizes.The experiment evaluates practical computational scaling of the MPS algorithm.

F. Arithmetic intensity

The algorithm’s dominant nonlinear matrix multiplications have high arithmetic intensity, making efficient use of modern parallel hardware possible. Memory efficiency remains important because poor cache utilization can invalidate the expected scaling.

F. Arithmetic intensity: Arithmetic intensity I is the ratio of performed FLOPs to required memory traffic, indicating whether hardware performance is compute- or memory-bandwidth-limited.Higher I generally enables more effective use of modern hardware.
F. Arithmetic intensity: The nonlinear term repeatedly performs matrix-matrix multiplications between non-square matrices whose dimensions depend on the bond dimension χ.These operations dominate the algorithm in practice.
F. Arithmetic intensity: Poor hardware or software memory utilization, including inadequate cache size or missing cache blocking, can prevent the theoretical arithmetic-intensity scaling from holding.The stated scaling assumes memory is efficiently utilized.
F. Arithmetic intensity: The dominant matrix-multiplication operation has high arithmetic intensity and can efficiently exploit GPUs, TPUs, and other parallelized computing hardware.This conclusion follows from the availability of highly optimized hardware and software for matrix multiplication.

A. Variational quantum algorithm

The quantum algorithm encodes the flow solution in a parameterized quantum state and minimizes a problem-dependent cost function using variational optimization. Its variational manifold can be more expressive than the classical MPS manifold and improves bond-dimension scaling even when restricted to MPS.

A. Variational quantum algorithm: The quantum algorithm encodes the solution in the probability amplitudes of a parameterized quantum state created from an initial state by a quantum-gate network.Problem-dependent gates and measurements evaluate a cost function.
A. Variational quantum algorithm: Variational optimization minimizes a quantum cost function whose trial state represents the current time-step solution and whose reference state is the previous solution.Gradient-based optimizers require only selected scalar products involving the trial state.
A. Variational quantum algorithm: The gradient calculation uses the real part of scalar products involving the trial state and tracks changing normalization through auxiliary parameters.The supplied passage identifies the adjoint operation and normalization-tracking parameters but does not provide the complete displayed equation.
A. Variational quantum algorithm: Two copies of the previous solution state allow the nonlinear term to be handled directly in the quantum circuit.The central circuit for this operation is illustrated in Supplementary Figure 5.
A. Variational quantum algorithm: The variational states are not limited to MPS, allowing a more expressive manifold and reducing bond-dimension scaling from O(χ4) to O(χ2) within the MPS manifold.The broader manifold is identified as having potential to lead to a quantum advantage.

B. Comparison to alternative proposals

The proposed quantum strategy shares logarithmic qubit scaling with alternative approaches but targets general nonlinear problems rather than efficiency in the number of time steps. The supplied supplementary figure passages mainly identify tensor-contraction illustrations without stating an outcome comparison.

B. Comparison to alternative proposals: Logarithmic scaling in the number of qubits with grid points gives all three quantum strategies potential for exponential speedup over some standard classical methods.This is the shared computational property distinguishing the proposals from the comparison perspective.
B. Comparison to alternative proposals: The proposed approach focuses on solving general nonlinear problems, whereas the alternatives focus on being efficient in the number of time steps.This is the main stated difference among the three strategies.
B. Comparison to alternative proposals: Supplementary Figure 6 illustrates tensor contractions between tensors, with contractions performed by summing over a closed bond while open bonds remain unsummed.The supplied caption identifies the figure as a tensor-contraction illustration but does not state a performance comparison.

6. GRAPHICAL NOTATION

Graphical tensor-network notation represents contractions and network connectivity more clearly than algebraic descriptions for complex geometries. The section contrasts MPS, TTN, and MERA according to which sites they connect and whether loops are present.

6. GRAPHICAL NOTATION: Graphical notation is introduced to describe MPS, TTNs, and MERA when algebraic descriptions become impractical for complicated geometries.The notation is intended to make tensor-network structures more legible.
6. GRAPHICAL NOTATION: Matrix multiplication between neighboring MPS tensors is equivalent to summing over an internal index, represented graphically as a tensor contraction.The same notation is used to draw the MPS decomposition.
6. GRAPHICAL NOTATION: MPS connects only neighboring sites, while TTN connects distant sites with fewer direct neighboring connections.The two geometries therefore encode different connectivity patterns.
6. GRAPHICAL NOTATION: MERA generalizes TTN by retaining nearby-site connectivity while adding greater connectivity and loops.The supplied figure passage identifies these features in the MERA decomposition.