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Simulating Quantum Turbulence with Matrix Product States

T0 review · 3 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A matrix product state solver for the damped Gross-Pitaevskii equation reproduces nonlinear excitations and turbulent statistics with 10x to over 10,000x memory compression.

desk verdict A solid quantics-MPS benchmark for the GP equation with convincing 1D/2D/3D validation, but the flagship reconnection and the 10,000x memory claim rest on self-convergence and unquantified extrapolation rather than a DNS baseline. read the letter →

arxiv 2508.12191 v2 pith:5ZIMU2VW submitted 2025-08-17 quant-ph physics.comp-ph

Felipe Gómez-Lozada , Nicolas Perico-García , Nikita Gourianov , Hayder Salman , Juan José Mendoza-Arenas This is my paper · ORCID
classification quant-phphysics.comp-ph
keywords quantumturbulenceGross-Pitaevskiiequationmatrixproductstatesquanticstensortraintime-dependentvariationalprinciplevortexreconnectionenergyspectrumnetworks
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the damped Gross-Pitaevskii equation, a standard model for Bose-Einstein condensates and quantum turbulence, can be time-evolved in a quantics matrix product state representation with dramatic memory savings. Because the solver retains only the strongest correlations between length scales, it uses 10x to over 10,000x less memory than direct numerical simulation while still reproducing dark solitons, vortex dipoles and rings, vortex reconnections, Kelvin waves, and turbulent energy spectra. If this holds, quantum turbulence simulations that previously required enormous grids become feasible on a single GPU, opening larger system sizes and longer evolution times. The paper supports the claim with infidelities below $10^{-4}$ for benchmark excitations and with statistical agreement for turbulent states in one, two, and three dimensions.

What carries the argument

The central object is the quantics MPS: a continuous function on a $2^N$ grid is reshaped into an $N$-qubit state whose qubits label binary length scales, then compressed by singular-value truncation. The paper's `staircase` ordering couples the same length scales of different axes and concentrates correlations at the center of the chain, which gives lower truncation error than interleaved or sequential orderings. Time evolution uses a two-site time-dependent variational principle with Strang splitting of the kinetic and nonlinear potential terms, where the nonlinear term is applied through a matrix product operator representing the Hadamard product. Derivatives use eighth-order finite differences built from shift-operator MPOs, and the kinetic term can also be applied via a low-rank quantum Fourier transform MPO. Vortex lines are extracted directly in the compressed representation using MPS sampling and Fourier interpolation, avoiding a costly full contraction. The method's cost is dominated by the $O(N\chi^4)$ Hadamard-product step.

What would settle it

A direct check would be to run the same MPS solver on a 3D turbulent state at a higher vortex line density or for a longer time and compare vortex density decay and the incompressible spectrum with DNS: if the required bond dimension grows faster than the square root of the grid-point count, or if the memory percentage rises steeply with system size at fixed density, the compression claim fails. For the flagship reconnection, comparing Kelvin-wave ring emission times against a DNS on a sufficiently large supercomputer would settle whether the self-convergence tests missed physics.

Watch

Extended reading notes

Core claim

The central discovery is that the wavefunction of the damped Gross-Pitaevskii equation, when encoded in a quantics binary length-scale basis, has a strongly decaying correlation structure that a matrix product state can exploit: truncating weak interlength-scale correlations yields accurate dynamics at a fraction of the direct-simulation cost. For single dark solitons, vortex dipoles, and vortex rings, MPS evolutions match DNS with infidelities near or below $10^{-4}$, using as little as 0.4% of the memory for 3D rings. For the flagship 3D vortex-line reconnection in a box of size $L=512\xi$, the method uses 0.03% of DNS memory, about 40 MB per snapshot, and still captures reconnections, Kelvin waves, the Crow instability, and vortex ring cascades. For turbulent states, the maximum bond dimension scales as the square root of the number of grid points, while the memory percentage depends mainly on the soliton or vortex density; the incompressible kinetic energy spectrum, including the $k^{-5/3}$ and $k^{-3}$ regimes, is recovered with modest bond dimensions.

Load-bearing premise

The method's utility rests on the empirical fact that the damped Gross-Pitaevskii wavefunction stays sufficiently low-entangled in the quantics length-scale encoding during turbulent evolution, so truncating virtual bonds at a few hundred retains the physically correct dynamics and statistics.

Editorial extensions

If this is right

  • If the central claim holds, 3D quantum turbulence simulations at vortex line densities around $2\times 10^{-5}\,\xi^{-2}$ and $L=512\xi$ need only tens of megabytes per snapshot rather than roughly 128 GB, fitting on a single 40 GB GPU.
  • Turbulent statistics such as two-point correlations, vortex density decay, and the incompressible energy spectrum are recoverable at bond dimensions below those needed for pointwise chaotic accuracy, so spectral studies can run even more cheaply.
  • The memory percentage depending on excitation density rather than system size implies that larger boxes at fixed density do not erase the compression advantage.
  • The same MPS pipeline extends to generalized Ginzburg-Landau models, dipolar and supersolid condensates, and other nonlinear multiscale PDEs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • In my reading, the near-constant memory percentage with system size is the load-bearing extrapolation: if a few percent of DNS memory holds at $2048^3$ grids, previously inaccessible sizes become reachable, but the paper does not derive this density-only dependence from the equations.
  • The success at recovering the incompressible spectrum with low bond dimensions suggests a general principle: statistics dominated by topologically protected vortex cores are cheaper to compress than full chaotic fields, which may transfer to other vortex-dominated turbulent systems.
  • A natural test is to force the system continuously and check whether the long-time statistically steady state remains low-entangled; the paper demonstrates decaying turbulence, and forced turbulence could generate more compressible sound-wave entanglement.
  • Replacing the Hadamard product with tensor cross interpolation would lower the nonlinear step from $O(N\chi^4)$ to $O(N\chi^3)$, which the paper identifies as a bottleneck; if successful, the practical speedup may exceed the memory savings.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript presents a quantics matrix product state (MPS) solver for the damped Gross-Pitaevskii equation, using a binary (quantics) encoding of spatial length scales, staircase ordering of qubits, two-site TDVP time evolution with Strang splitting, and MPO representations of finite-difference derivatives. The authors benchmark the solver against DNS and analytic solutions for a 1D dark soliton, a 2D vortex dipole, and a 3D vortex ring, reporting infidelities of order 1e-4 or below. They then apply the method to a 2048^3 vortex-line reconnection in a 512ξ box, to a 1D soliton gas, to 2D vortex turbulence, and to a 3D vortex tangle, reporting that memory usage can be reduced by factors of 10 to more than 10,000 relative to DNS while reproducing vortex densities, correlation functions, and energy spectra above a threshold bond dimension. A scaling analysis in Sec. V A claims that the required bond dimension grows as ~sqrt(L^d) for fixed excitation density while the memory percentage saturates to a density-dependent constant.

Significance. The central methodological claim—that quantics MPS time evolution can efficiently compress the GP wavefunction across the dynamically relevant length scales—is well supported for the benchmarked single-excitation systems. The paper's careful comparison with independent DNS and analytic solutions, with no fitted parameters, gives the 1D/2D/3D excitation results high credibility. The statistical comparisons for turbulence (vortex density decay, two-point correlation functions, and kinetic energy spectra) also provide concrete, falsifiable benchmarks that will be useful to the community. If the 512ξ reconnection and the memory-scaling extrapolation can be validated more strongly, the work would be an important advance for quantum turbulence simulation, enabling system sizes that are currently prohibitive for DNS. The main limitations are that the flagship reconnection lacks an external DNS baseline and that the central 'over 10,000x' memory claim rests on an unquantified extrapolation in Fig. 10.

major comments (3)
  1. [Sec. IV D, Appendix E, Figs. 6–9 and 25] The >10,000x memory-reduction demonstration rests solely on the 512ξ reconnection, for which the only validation is self-convergence in bond dimension. Because the two runs share the same quantics encoding, finite-difference derivative, Strang splitting, and TDVP manifold projection, mutual agreement cannot exclude a systematic truncation bias in reconnection physics. I request an external validation at a scale accessible to DNS, for example the same four-vortex setup in a smaller periodic domain (e.g., L=64ξ or 128ξ), comparing reconnection time, emitted vortex ring radii, and Kelvin wave dynamics between MPS and split-step DNS. If such a DNS comparison is not feasible, the abstract's 'over 10,000x' claim should be weakened to the cases actually benchmarked against DNS.
  2. [Sec. V A, Fig. 10(j–l)] The claim that the memory percentage saturates to a constant at fixed excitation density, which is used to extrapolate to the 5760^3 comparison and to support the largest memory-reduction factors, is asserted without derivation and without statistical error bars. The six-state averages in panels (a–c) are not propagated to panels (j–l), and the linear interpolation used to obtain the fixed-density curves is not accompanied by residuals or fit uncertainties. Please add the spread across random initial states, a statistical characterization of the saturation, and either a derivation of the constant-memory scaling or an explicit statement that the extrapolation is conjectural.
  3. [Appendix E, Fig. 25(b,c)] The relative y-momentum error (≈1e-3) is four orders of magnitude larger than the energy error (≈1e-7), yet the text describes both as showing that the conserved quantities are 'well conserved.' Because momentum conservation is part of the validation of the reconnection simulation, this discrepancy needs an explanation (for example, accumulation of phase error in the finite-difference derivative, truncation of high-frequency components, or a periodic-boundary effect), together with reported conservation of P_x and P_z, before the self-convergence argument can be considered conclusive.
minor comments (6)
  1. [Sec. III A] There is a typo: 'choosen' should be 'chosen'.
  2. [Sec. IV B] There are typos: 'anhihiated' should be 'annihilated' and 'anihilation' should be 'annihilation'.
  3. [Sec. V B 2] The 'Mandelung transformation' should be the 'Madelung transformation.'
  4. [Sec. IV D vs Appendix E] The main text says the convergence tests confirm that results vary by less than 10^-4 in infidelity when the bond dimension is increased, but Fig. 25(a) reports I≤10^-5 for the largest bond-dimension step; these numbers should be reconciled.
  5. [Sec. VI] The statement 'memory reduction of 4 orders of magnitude (0.03%)' is arithmetically inconsistent: 0.03% corresponds to about 3.3 orders of magnitude, not 4. The earlier 0.002% figure is closer to 4.3 orders, so the wording should be adjusted to match the actual compression ratio used.
  6. [Fig. 10] The dashed power-law lines labeled ~sqrt(L), ~L, and ~L^{3/2} would be more informative if the fitted exponents and confidence intervals were reported; this is a presentation issue but would strengthen the scaling claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the MPS method is benchmarked against independent DNS and analytic solutions; the reconnection run uses self-convergence, which is a validation limitation rather than a circular derivation.

full rationale

I walked the derivation chain: the central quantitative claims are the memory ratios (defined arithmetically by Eq. (2) from measured bond dimensions), the benchmark infidelities against DNS (Figs. 3–5, 11–16), and the reproduction of two-point correlations and energy spectra in 2D turbulence (Figs. 12–14). None of these involves fitting a parameter to a subset of data and then predicting that same subset. The soliton test is checked against the analytic Jacobi-elliptic solution (Appendix B1), and the 2D/3D benchmarks are checked against split-step Fourier DNS. The scaling chi_max ~ sqrt(L^d) is presented as an observed trend and traced back to the untruncated SVD bound chi_SVD_max = sqrt(M); this is a consistency statement, not a self-imported constraint. The only self-referential piece is the 512-xi reconnection validation, where the paper states: "Due to the large grid size we employed to model this physics, it is impractical to use DNS... instead of comparing to the respective DNS, alternative convergence criteria are discussed in Appendix E." Appendix E compares MPS runs of different bond dimensions and checks energy/momentum conservation. That is a legitimate numerical convergence check, not an equation that reduces to its own input; it is weaker external evidence than a DNS comparison, but it does not make the memory-reduction or reconnection-topology claims circular. Self-citations (e.g., Gourianov et al. for quantics turbulence) are background and are not load-bearing: the benchmarks here are self-contained against external DNS/analytic references. Overall, no circular step meets the bar of Eq. X = Eq. Y by construction or fitted-input-renamed-as-prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The method rests on standard tensor-network mathematics plus domain assumptions about the GP model, the representativeness of the initial ensembles, and the persistence of low entanglement during turbulent evolution. No new physical entities or fitted constants are introduced; the computational parameters chi_max and tolerances are numerical convergence controls, not fitted physical parameters.

assumptions (6)
  • domain assumption The damped Gross-Pitaevskii equation is a valid mean-field model for the Bose-Einstein condensate dynamics studied here.
    Invoked in Sec. II; the paper uses it as the equation of motion, and its validity for the regimes simulated is taken from prior cold-atom literature.
  • standard math The quantics binary encoding (Eq. (3)) maps continuous functions to tensors such that bounded interlength-scale correlations imply low bond dimension.
    Foundational for the method (Sec. III A; established in Refs. [47,48]).
  • domain assumption The time-dependent variational principle with Strang splitting converges to the exact GP evolution when bond dimension is sufficient.
    Used throughout Sec. III B; the paper validates this empirically but it is not proven for the nonlinear, damped GP equation.
  • domain assumption A random phase interpolation with imaginary time evolution generates representative turbulent initial states with specified vortex/soliton densities.
    Sec. V A and Appendix B 5; the representativeness of these ensembles is asserted, not proven.
  • domain assumption The eighth-order finite difference Laplacian in the MPO representation is accurate at the xi/4 grid resolution used.
    Appendix A 2; accuracy is benchmarked only indirectly through infidelity comparisons.
  • domain assumption Truncated bond dimensions introduce errors small enough that statistical quantities of turbulence are preserved, even when pointwise wavefunctions diverge.
    Sec. V B; the paper demonstrates this for chosen densities and times, but it is the key empirical premise.

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Pith. "Pith review of Simulating Quantum Turbulence with Matrix Product States." pith.science (2026). https://pith.science/paper/5ZIMU2VW

@misc{pith2026250812191,
  author       = {Pith},
  title        = {Pith review of: Simulating Quantum Turbulence with Matrix Product States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ZIMU2VW}},
  note         = {Machine review of arXiv:2508.12191}
}
abstract

Quantum turbulence spans length scales from the system size $L$ to the healing length $\xi$, making direct numerical simulations (DNS) of the Gross-Pitaevskii (GP) equation computationally expensive when $L \gg \xi$. We present a matrix product state (MPS) solver for the GP equation that efficiently compresses the wavefunction by truncating weak interlength-scale correlations. This approach reduces memory usage by factors ranging from 10x to over 10,000x compared to DNS. We benchmark our approach on nonlinear excitations, namely dark solitons (1D) and quantized vortices (2D, 3D), capturing key dynamics such as Kelvin wave propagation and vortex ring emission in the case of vortex line reconnection. For turbulent states composed of multiple nonlinear excitations, we find that the memory compression of the MPS representation is directly proportional to the soliton or vortex densities. We also accurately reproduce established results from two-point correlation functions and energy spectra, where we recover the incompressible kinetic energy spectrum with little memory overhead. These results demonstrate the representative capabilities of the MPS ansatz for quantum turbulence and pave the way for studying this nonequilibrium state using previously-prohibited system sizes to uncover novel physics.

Figures

Figures reproduced from arXiv: 2508.12191 by the authors.

Figure 1
Figure 1. FIG. 1. Examples of tensor network diagrams. (a) Tensors [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Example of MPS orderings of 2D (3D) functions in a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Evolution of a dark soliton. (a) Analytical density [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Evolution of a vortex dipole. (a-i) Density profile [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Evolution of a vortex ring. (a-f) Density contours [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Vortex reconnection of perpendicular dipoles. We [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Generation of Kelvin waves after the reconnection [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Vortex line collapse due to perturbations from the reconnection shown in Fig. [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Density profile [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Truncation parameters for the MPS representation of initial turbulent states created using DNS with an infidelity of [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Evolution of soliton gas with initial density [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Density profiles of the DNS (a) and MPS simulation [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. (a) Vortex density [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Time-averaged scaled kinetic energy spectrum [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. Vortex density profiles of a 3D quantum turbulent state for [PITH_FULL_IMAGE:figures/full_fig_p015_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Vortex density [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. Depiction of the Hadamard product. (a) Promoting [PITH_FULL_IMAGE:figures/full_fig_p017_17.png]
Figure 19
Figure 19. Figure 19: FIG. 19. Density profile [PITH_FULL_IMAGE:figures/full_fig_p018_19.png]
Figure 18
Figure 18. Figure 18: FIG. 18. Density profile [PITH_FULL_IMAGE:figures/full_fig_p018_18.png]
Figure 20
Figure 20. Figure 20: FIG. 20. Initial condition for the vortex ring with periodic [PITH_FULL_IMAGE:figures/full_fig_p019_20.png]
Figure 21
Figure 21. Figure 21: FIG. 21. 3D extension of 2D quantics MPS. The yellow and [PITH_FULL_IMAGE:figures/full_fig_p019_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. Density profiles [PITH_FULL_IMAGE:figures/full_fig_p020_22.png]
Figure 23
Figure 23. Figure 23: FIG. 23. Comparison of vortex dipole quantics MPS represen [PITH_FULL_IMAGE:figures/full_fig_p021_23.png]
Figure 24
Figure 24. Figure 24: FIG. 24. Two-site mutual information [PITH_FULL_IMAGE:figures/full_fig_p022_24.png]
Figure 25
Figure 25. Figure 25: FIG. 25. Convergence tests of the MPS results from Figs. [PITH_FULL_IMAGE:figures/full_fig_p024_25.png]

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Reference graph

Works this paper leans on

191 extracted references · 54 canonical work pages

  1. [1]

    Hadamard product Consider an MPS that represents a function using the quantics framework from Sec. III A. The element- wise multiplication operation for this format is called the Hadamard product [51], which is performed using 3-rank delta tensorsδi,j,k that equal 1 fori=j=kand zero oth- erwise. We apply a delta tensor for each physical index in the MPS, ...

  2. [2]

    shoulder

    2D Point-vortex distribution We choose a system size ofL= 512ξwith a vortex den- sity ofρ 2D Vortex = 4.2×10 −3ξ−2. We let the state evolve fort= 200ξ/cwithγ= 10 −2. The density profiles of the final states are shown in Fig. 12, for DNS (panel (a)) and MPS withχ max = 360 (panel (b)). Here, we observe similar vortex distributions with small discrepancies....

  3. [3]

    We consider an initial state with a density ofρ 3D Vortex = 1.1×10 −2ξ−2 in a domain withL= 64ξwith the fixed resolution ξ/4

    3D Vortex tangle Finally, we simulate the dynamics of a 3D turbulent system using the MPS time evolution. We consider an initial state with a density ofρ 3D Vortex = 1.1×10 −2ξ−2 in a domain withL= 64ξwith the fixed resolution ξ/4. We evolve the turbulent state untilt= 128ξ/c withγ= 10 −2. We show snapshot samples in Fig. 15 for DNS (upper row) and TDVP (...

  4. [4]

    To build the latter, we first write 1D 2D 3D Sequential 3 4 4 Staircase 4 4 Interleaved 6 8 TABLE I

    Derivatives The derivatives are constructed using an eighth-order finite difference approximation with periodic boundary conditions [160]. To build the latter, we first write 1D 2D 3D Sequential 3 4 4 Staircase 4 4 Interleaved 6 8 TABLE I. Bond dimensionsχof the Laplacian∇ 2 MPO rep- resentation in one to three dimensions using the sequential, staircase, ...

  5. [5]

    1D Dark soliton Using the exact result from Ref. [98] we build the dark soliton ansatzψ(x) = p ρ(x)eiϕ(x) corresponding to ρ(x) =ρ 1 + (ρ0−ρ 1)cn u x L;k ,k 2 ,(B1) ϕ(x) =vx− Lv 2 +π Π h 1− ρ1 ρ0 ,am u x L;k ,k ,k i Π h 1− ρ1 ρ0 ,k i . (B2) Herecn(x,k) andam(x,k) are the elliptic cosine and Ja- cobi amplitude, Π(n,k) and Π(n;ϕ|k) are the complete and inco...

  6. [6]

    The vortex dipole is characterized by a dipole lengthd=|r 2−r 1|, andθ j(r) = Angle(r−r j,r 2−r 1) is the angle ofr−r j measured from the directionr 2−r 1

    2D V ortex dipole Here, the wave function consists of the composition of an ansatz for each vortex, both with opposite charges, given by ψ(r) =ψ 1(r)·ψ ∗ 2(r)·e i(θ1(x,L)−θ2(x,L)) 2y L,(B5) ψj(r) = s |r−r j|2(a1 +a 2|r−r j|2) 1 +b 1|r−r j|2 +b 2|r−r j|4 eiθj(r),(B6) a1 = 11/32,(B7) a2 =a 1(b1−1/4),(B8) b1 = 5−32a 1 48−192a 1 ,(B9) b2 =a 2,(B10) wherer= (x...

  7. [7]

    Then, for a given ra- diusRwe definer 1 = (−R,L/2),r 2 = (R,L/2) and r= ( p (x−L/2) 2 + (y−L/2) 2,z), and substitute in 19 FIG

    3D V ortex ring The vortex ring is generated by taking the ansatz from Appendix B 2 and rotating it with respect to the axis along which the dipole moves. Then, for a given ra- diusRwe definer 1 = (−R,L/2),r 2 = (R,L/2) and r= ( p (x−L/2) 2 + (y−L/2) 2,z), and substitute in 19 FIG. 20. Initial condition for the vortex ring with periodic boundary condition...

  8. [8]

    Also, an imaginary time evolution is ap- plied to suppress initial sound waves

    V ortex line reconnection Each dipole is initialized in a 2D quantics MPS by us- ing the initial condition from Appendix B 2 and applying iterative SVDs. Also, an imaginary time evolution is ap- plied to suppress initial sound waves. The 2D quantics MPS is then extended to a 3D quantics MPS by adding delta tensors for the missing qubits that represent the...

Show all 191 references
  1. [9]

    For the DNS, we sample uniformly from the inter- val [−1,1] for both the real and imaginary components

    Random phase interpolation Motivated by Kobayashi,et al.[115], we first create a coarse grid withN RPI < Nqubits per axis, choosing randomly-generated values of the wave function on each point. For the DNS, we sample uniformly from the inter- val [−1,1] for both the real and i...

  2. [10]

    Afterwards, we create a semi-diagonal tensor connected to both bonds, defined with an identity state as|0⟩+|1⟩forl=rand zero oth- erwise

    On the other hand, if it has to be added in the mid- dle of the chain, we split the corresponding bond index cinto two equivalent copiesl,r. Afterwards, we create a semi-diagonal tensor connected to both bonds, defined with an identity state as|0⟩+|1⟩forl=rand zero oth- erwise...

  3. [11]

    2D V ortex dipole We compare in Fig. 23 the density profiles of a vortex dipole after truncating their MPS representation with the interleaved (a) and staircase (b) orderings, the two possi- bilities considered in this work for 2D. We useχ max = 8. Here we observe that, for th...

  4. [12]

    23, we focus our anal- ysis on the three-dimensional staircase ordering shown in Fig

    3D V ortex ring Motivated by the results in Fig. 23, we focus our anal- ysis on the three-dimensional staircase ordering shown in Fig. 2(d). This MPS ordering has six different pos- sibilities indicated by the various possible permutations of the axes (x,y, orz) across the MPS...

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    General algorithm To obtain a single vortex line, we follow the next steps:

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    Choose an initial point in the gridr 0. The advan- tage of the MPS ansatz is that it can be interpreted as a probability distribution of its free index entries, for which efficient sampling algorithms have been developed [157, 169]. In particular, points close to a vortex line...

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    The first vector (ˆ w0) points in the direction of the pseudovorticityω 0 evaluated at the initial sampled point; this indicates the di- rection of the vortex line

    Define a set of orthonormal vectors at the initial sampled pointr 0. The first vector (ˆ w0) points in the direction of the pseudovorticityω 0 evaluated at the initial sampled point; this indicates the di- rection of the vortex line. The other two (ˆ u 0,ˆ v0) are arbitrary ve...

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    ∆urn ∆vrn # =−  Re ∂(n) u ψn Re ∂(n) v ψn Im ∂(n) u ψn Im ∂(n) v ψn   −1

    Apply a Newton–Raphson method to find the clos- est zero ofψfrom the initial sampled pointr 0 in the plane perpendicular to the vortex line. The vortex core estimationr n at then-th iteration is obtained according tor n+1 =r n + (∆urn)ˆ un + (∆vrn)ˆ vn, where " ∆urn ∆vrn # =− ...

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    This evolution is performed so that the next sampled point follows the direction of the vortex line given byˆ w n atr n

    To find the coordinates of the subsequent point in the vortex line, the following initial point is chosen asr n+1 =r n +ζˆ wn whereζis the resolution of the vortex line. This evolution is performed so that the next sampled point follows the direction of the vortex line given b...

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    This method is iterated to obtain the complete set of vortex lines

    The previous steps are repeated until the lastr n 23 and firstr 0 vortex core positions fullfill|r n−r 0|< ζ, indicating the closing of the vortex line. This method is iterated to obtain the complete set of vortex lines. To avoid double counting, Ref. [106] pro- posed using a ...

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    F ourier interpolation To efficiently recover the information of the wavefunc- tionψand its derivatives∇ψat an arbitrary positionr n (in then-th iteration of the Newton-Raphson step from Appendix D 1) with subgrid resolution, we use Fourier interpolation [170]. The procedure c...

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    III B, the largest computational costs scale asO(χ 4) and correspond to the Hadamard products used to cal- culate|ω| 2

    Computational cost In analogy to the time evolution method presented in Sec. III B, the largest computational costs scale asO(χ 4) and correspond to the Hadamard products used to cal- culate|ω| 2. Then, the calculation of the MPSsF[ψ] and F[∇ψ] each scales asO(χ 3) and only ha...

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