A Quantum Inspired Approach to Exploit Turbulence Structures

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.Source: RESULTS · p. 2
• 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.Source: RESULTS · p. 2
• 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.Source: RESULTS · p. 3

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.Source: RESULTS · p. 4

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.Source: RESULTS · p. 4

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.Source: RESULTS · p. 5

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.Source: DISCUSSION · p. 6

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.Source: DISCUSSION · p. 6

• The paper lists eight authors, including Nikita Gourianov and Dieter Jaksch.Source: Supplementary Information: A Quantum… · p. 12
• The authors are affiliated with institutions in the United Kingdom, United States, Singapore, and Germany.Source: Supplementary Information: A Quantum… · p. 12
• The work is identified as arXiv:2106.05782v3, dated 4 Jul 2022.Source: Supplementary Information: A Quantum… · p. 12

• 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.Source: §1 · p. 14

• The supplementary spectra include d99(n, t) contours to assess whether truncation discards relevant interscale correlations.Source: A. Schmidt spectra · p. 14
• 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.Source: Figure 1 · p. 14
• The Schmidt coefficients are sorted in descending order and normalized so that their squares sum to 1.Source: A. Schmidt spectra · pp. 14–15
• The black dashed lines mark d99(n, t), the truncation values used in the main text.Source: A. Schmidt spectra · p. 14
• 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.Source: Figure 2 · p. 15

B. Entanglement entropy

• Von Neumann entanglement entropy is computed from the Schmidt spectrum λ_α(n, t).Source: B. Entanglement entropy · p. 16
• In the TDJ, entropy shifts toward lower n over time, consistent with an inverse energy cascade toward coarser length scales.Source: B. Entanglement entropy · p. 16
• In the TGV, entropy increases at larger n over time, consistent with a direct cascade toward finer length scales.Source: B. Entanglement entropy · p. 17
• The TGV entropy increases with time across all bipartitions, accompanying disorder from collapse into worm-like vortical structures.Source: B. Entanglement entropy · p. 17

MPS encoding

• The grid contains M = 2^(KN) points, and binary index bits are grouped by length scale to organize the flow representation.Source: A. Encoding a flow component as a… · p. 18
• An MPS approximates each discretized velocity component using matrices A^ω_n whose bond dimension χ controls the captured interscale correlations.Source: A. Encoding a flow component as a… · p. 18, C. Schmidt decomposition · p. 19
• At maximal χ, the MPS becomes exact and uses Q = 2^(KN) = M parameters after accounting for gauge degrees of freedom.Source: A. Encoding a flow component as a… · p. 18, B. Number of physical variables · p. 18

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.Source: E. Encoding quantum states · p. 20

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.Source: B. Mathematical solution for initial δ… · pp. 22–23
• 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.Source: B. Mathematical solution for initial δ… · p. 23
• 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.Source: B. Mathematical solution for initial δ… · p. 24

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.Source: C. Matrix product state representation… · p. 24
• 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.Source: C. Matrix product state representation… · p. 24, D. Triangular waves as matrix product… · p. 24
• 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.Source: C. Matrix product state representation… · p. 24
• 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.Source: D. Triangular waves as matrix product… · pp. 24–26
• 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.Source: §4 · p. 27, A. The variational problem · p. 28, B. Local minimisation · pp. 28–29

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.Source: D. Theoretical computational scaling · p. 29
• 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.Source: Figure 4 · p. 30, D. Theoretical computational scaling · p. 30
• 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.Source: D. Theoretical computational scaling · pp. 29–30
• 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.Source: D. Theoretical computational scaling · p. 30

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.Source: E. Demonstration of computational… · p. 30

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.Source: F. Arithmetic intensity · pp. 30–31
• 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.Source: F. Arithmetic intensity · p. 31
• 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.Source: F. Arithmetic intensity · p. 31
• 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.Source: F. Arithmetic intensity · p. 31

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.Source: A. Variational quantum algorithm · p. 32
• 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.Source: A. Variational quantum algorithm · p. 32
• 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.Source: A. Variational quantum algorithm · p. 32
• 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.Source: A. Variational quantum algorithm · p. 32
• 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.Source: A. Variational quantum algorithm · p. 32

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.Source: B. Comparison to alternative proposals · p. 32
• 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.Source: B. Comparison to alternative proposals · p. 32
• 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.Source: Figure 6 · p. 34, B. Comparison to alternative proposals · p. 34

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.Source: §6 · p. 34
• 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.Source: §6 · p. 34
• MPS connects only neighboring sites, while TTN connects distant sites with fewer direct neighboring connections.The two geometries therefore encode different connectivity patterns.Source: §6 · p. 34, Figure 7 · p. 35
• 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.Source: §6 · pp. 34–35