PaperlayerView exact revision on arXiv

Source-linked AI summary

Scalable quantum simulation of continuous-time generative models via tensor networks

Nathan X. Kodama, L. Andrew Wray, Sam Cochran, Chad Rigetti, Shravan Veerapaneni, Michael J. Keiser

arXiv:2608.21700v1quant-phcs.AIcs.LG

TL;DR

Statistical inference from trained continuous-time generative models is costly, motivating scalable ways to simulate their learned transport. This paper uses tensor networks for wavefunction flows, converting classical simulation costs into polynomial-in-dimension growth and reproducing idealized rare-event amplitude-amplification scaling.

  • Problem

    Statistical properties of trained continuous-time generative models are costly to extract because expectations and tests require repeated sampling.

  • Method

    The paper uses tensor networks to compress, evolve, and represent wavefunctions while querying scalar-valued potentials pointwise instead of tabulating dense grids.

  • Results

    Tensor-network compression converts classical wavefunction-flow simulation costs into polynomial-in-dimension growth and reproduces idealized O(1/√p_rare) rare-event scaling.

  • Takeaways & Limitations

    The pipeline delivers a wavefunction-flow state on which amplitude amplification reproduces idealized rare-event sampling costs, using 2.5× fewer preparations than rejection sampling at the same threshold.

  • Takeaways & Limitations

    Smoothness alone does not ensure small bond dimension, and correlated target geometry can make bond dimension the binding constraint at high dimension.

Abstract

from arXiv · show

Continuous-time flow and diffusion models are widely used across many application domains, from large-scale deployment in computer vision and protein folding to emerging adoption for modeling language, time series, and quantum states. After training, inferring statistical properties from continuous-time models is costly. Wavefunction flows target this cost by recasting learned transport as unitary evolution, whose final Born distribution approximates the target distribution. This prepares a coherent amplitude encoding (a qsample) that can be post-processed by quantum algorithms offering a quadratic advantage over Monte Carlo sampling. We present the first numerical study of these flows, in which we represent time-dependent potentials and states as tensor networks. At spatial dimension $d=8$, storage falls by $\sim 10^7\times$ relative to the dense grid of $N^d$ points, and evolution wall-clock time falls by $\gtrsim 10^3\times$ against a baseline extrapolated from the measured $d\le 5$ scaling. We validate our pipeline by reproducing the $O(1/\sqrt{p_{\rm rare}})$ scaling of rare-event sampling.

1 Introduction

Wavefunction flows offer a quantum route to extracting statistics from continuous-time generative models, but their numerical study is hindered by exponentially scaling dense-grid costs. This work introduces tensor-network simulations that make these flows tractable and reproduce dense-grid sample quality across dimensions two to eight.

  • Motivation: Continuous-time generative models transport probability through dynamics, but estimating expectations, moments, and property tests after training requires costly averaging.This motivates wavefunction flows as a route to more efficient statistical inference.
  • Challenge: Wavefunction flows had remained a purely theoretical proposal because direct numerical study was obstructed by exponential computational costs.The dense-grid potential step becomes especially expensive as spatial dimension increases.
  • Tensor-network method: MPS representation replaces the N^d dense grid with memory O(dND^2), using one rank-≤D core per spatial axis.The evolving wavefunction is compressed as a matrix product state, while the kinetic step remains site-local through one-dimensional Fourier transforms.
  • Quantum implementation: MPS wavefunctions are candidate inputs for sequential state preparation and can initialize parametrized quantum circuits to mitigate barren plateaus and provide warm starts.State preparation uses ancilla qubits logarithmic in bond dimension and a depth-linear sequence.
  • Contributions: ∼10^7× less memory and ≳10^3× less evolution time are achieved at eight dimensions while matching exact dense-grid simulation in sample quality.The comparison uses Gaussian mixture targets spanning two to eight dimensions and an evolution-time baseline extrapolated from measured scaling through dimension five.

2 Results

The results show that wavefunction flows reproduce target structure and predicted discretization-error rates while tensor-network compression enables substantial savings in memory and evolution time. Rare-event sampling follows the expected quadratic amplitude-amplification scaling, although the demonstration is idealized.

  • Error scaling: N −3/2 is the observed spatial/grid-error rate for the Laplace target, while the Trotter error follows global O(K−1) scaling.The spatial rate is fixed by the kink in the target amplitude, and the Trotter rate reflects accumulated local step errors.
  • Target reproduction: At every d ∈{3, 5, 7}, TDVP samples reproduce the target mode structure, resolve the same number of clusters, and achieve visually comparable occupancy.The comparison uses a panel-local t-SNE embedding and is intended qualitatively because t-SNE distorts global distances and densities.
  • Accuracy: At six of seven dimensions, TCI+1TDVP is at least as accurate as JAM; its worst case is a factor of 1.06 at d=8.Its best-cell SW values remain within a factor of 1.7 of the target–target reference across d = 2−8, without upward drift.
  • Scalability: At d=8, the TCI+1TDVP best cell uses approximately seven orders of magnitude less memory and at least three orders of magnitude less evolution wall-clock time than an extrapolated dense-grid baseline.Within the searched box, K = 64 and D = 32 at d = 8 are Pareto-optimal, with residual Trotter error O(1/K) still dominant.
  • Integrator comparison: χ⋆≈16 is the self-limiting bond for pure 2TDVP, whereas 1TDVP+SE must grow χ at larger K to suppress Trotter error.Pure 2TDVP reaches lower final sliced Wasserstein, and V-step accuracy is limited by Trotter-step count rather than bond capacity.
  • Rare-event sampling: O(1/√prare) state preparations per accepted rare sample replace O(1/prare) classical rejection-sampling draws under amplitude amplification.Both pipelines use the same learned potential, so the comparison isolates transport and sampling costs; the demonstration models an exact Grover outcome distribution rather than compiling Uprep.

3 Discussion

Tensor-network wavefunction flows replace exponential classical simulation costs with polynomial-in-dimension growth for empirically low-rank targets, while preserving a Born-density construction subject to discretization, Trotter, and compression errors. The simulations provide compact state and generator representations, but compiled quantum state preparation and qualifying readout tasks remain open challenges.

  • Scaling and compression: Tensor-network compression converts wavefunction-flow simulation from exponential cost to polynomial-in-dimension growth.The approach compresses, evolves, and represents wavefunctions while querying the scalar potential pointwise rather than tabulating it densely.
  • Scaling and compression: Empirically low-rank velocity potentials and evolving states place the demonstrated orthogonal Gaussian-mixture instances in the dequantizable regime.The target family is low-rank by construction, and quantum-inspired classical algorithms can match broad quantum speedups for low-rank, ℓ2-sampleable inputs with classical outputs.
  • Scaling and compression: Smooth wavefunctions, natural-scale potential structure, and phase action within the occupied subspace explain the observed compression.The wavefunction inherits regularity from the learned potential, cross interpolation discovers structure without external guidance, and the phase operator largely stays within the wavefunction’s occupied subspace.
  • Relationship to prior approaches: Wavefunction flows construct a Born density matching the target by construction, up to grid, Trotter, and compression errors.Unlike models that train a circuit or MPS to approximate a target, this pipeline simulates transport directly and occupies an intermediate position between quantum generative models and compiled-oracle simulations.
  • Limitations and open problems: Tensor-network simulations produce an MPS qsample but not yet a compiled unitary for coherent preparation on quantum hardware.A compact MPS admits sequential, logarithmic-depth unitary-circuit, or adaptive constant-depth constructions; choosing among them remains open.
  • Limitations and open problems: Classical simulability depends jointly on the state, generator, and readout ranks, so any advantage must specify a preparation-and-readout task.The relevant ranks are DΨ, DV, and DO; low bond growth alone does not establish classical difficulty or quantum advantage.

4 Methods … 4.3 The TCI→TDVP pipeline

The methods represent high-dimensional wavefunction flows with MPS tensor networks, split evolution into kinetic and potential steps, and implement the potential step through a TCI→TDVP pipeline that avoids dense-grid operators. The pipeline combines adaptive low-rank potential construction with variational MPS evolution while supporting fixed or adaptive bond dimensions.

  • 4 Methods: The methods comprise MPS representation, Trotter splitting, TCI→TDVP potential evolution, V-step integrators, pretrained velocity networks, and experimental protocols.The variational formalism underlying the integrators is derived separately in Supplementary Section S1.3.
  • 4.1 MPS representation and the scaling axis: O(dND^2) memory replaces N^d dense-grid storage because an MPS uses d sites of size N × D × D.The scaling study focuses primarily on spatial dimension d at moderate resolution, typically N ≤32.
  • 4.2 Trotter splitting: O(dND^2 log N) time and trivial memory characterize the MPS kinetic step, which applies one-dimensional FFTs site-locally because K is a Kronecker sum.The dense-grid implementation instead applies two d-dimensional FFTs per Trotter step.
  • 4.2 Trotter splitting: O(∆t^2) local error and O(K−1) global error arise from the eight-step commutator product formula used for each Trotter step.The split evolution separates kinetic K from the time-dependent diagonal potential V_t.
  • 4.3 The TCI→TDVP pipeline: TCI constructs a low-rank MPO for the generator V_t by adaptively sampling grid points, rather than forming or evaluating the potential on the full N^d grid.The method uses maxvol-style pivot selection and reconstructs the potential from the selected samples.
  • 4.3 The TCI→TDVP pipeline: 1TDVP keeps bond dimension fixed with one-site updates, whereas 2TDVP updates two sites and uses SVD truncation to Dmax for adaptive bond dimension.The passage states that 1TDVP offers the better tradeoff among wall-clock considerations, while 2TDVP requires a more expensive sweep.
  • 4.3 The TCI→TDVP pipeline: The pipeline avoids forming e^{iβV_t} directly because its bond dimension grows with β∥V_t∥, and instead evolves with the smoother generator V_t using TDVP.For the studied targets, the generator’s bond dimension remains empirically small and is generically far below that of its exponential.
  • 4.3 The TCI→TDVP pipeline: O(dND^2) state and O(dND^2 V) operator-construction sizes keep every touched object below dense-grid scale throughout alternating kinetic and V-steps.The flow repeats the V-step and kinetic step for K Trotter steps of the eight-letter product formula; pretrained MPS potentials can skip TCI.

4.4 Dense-grid pseudospectral baseline · 4.5 Tensor cross interpolation

The dense-grid pseudospectral baseline applies FFT-based kinetic evolution and pointwise potential multiplication but becomes infeasible at large grids. Tensor cross interpolation instead constructs compact MPS/MPO representations from selected function queries, exploiting smooth generator structure for efficient wavefunction evolution.

  • 4.4 Dense-grid pseudospectral baseline: 16 N^d bytes of complex128 memory makes the dense-grid baseline infeasible for N^d ≳ 10^9.The baseline applies Supplementary Eq. (S5) on the full N^d grid.
  • 4.4 Dense-grid pseudospectral baseline: Two d-dimensional FFTs implement e^{iαK}, while e^{iβV_t} is applied by pointwise multiplication.A sign mask centers the kinetic spectrum.
  • 4.5 Tensor cross interpolation: TCI constructs an MPS approximation of a d-variable discrete function without materializing the dense N^d tensor.It queries the function at a small set of carefully chosen pivot index tuples.
  • 4.5 Tensor cross interpolation: The MPS represents f(i_1, …, i_d) as a product of site tensors A_1[i_1]A_2[i_2] ··· A_d[i_d], with boundary dimensions D_0 = D_d = 1.TCI builds this representation site by site.
  • 4.5 Tensor cross interpolation: At each bond, TCI selects row and column index sets and builds a maxvol-optimized interpolating skeleton from the corresponding function block.DMRG-style sweeps refine the index sets to maximize the absolute determinant of the pivot block.
  • 4.5 Tensor cross interpolation: The pipeline applies TCI to the real generator V_t(x), producing a position-diagonal MPO of bond dimension D_V for TDVP wavefunction evolution.The MPO is diagonal in the position basis at each site.
  • 4.5 Tensor cross interpolation: Crossing smooth V_t rather than oscillatory e^{iβV_t} keeps ranks small in practice, because the exponential’s rank inflates with β∥V_t∥.The implementation exploits position diagonality to reduce per-site memory by a factor of N.
  • 4.5 Tensor cross interpolation: The cross is controlled by the maxvol residual and capped at D_V pivots over a fixed number of DMRG-like sweeps; crossing e^{iβV_t} instead would incur inflated rank D_exp.The build-then-apply alternative was not benchmarked.

4.6 1TDVP (one-site)

The one-site TDVP sweep integrates two local ODEs and uses adaptive Krylov matrix-exponential actions for site and bond updates. Because the bond dimension is preserved, 1TDVP is fixed-rank with per-Trotter-step cost O(d·N·D3·k), while retaining accuracy on the Gaussian mixture target.

  • Each single-site projector sweep update integrates two ODEs at every site j.
  • The effective site and bond operators are exponentiated for time ∆t using SciPy’s adaptive Krylov expm multiply.The Krylov subspace grows adaptively until the matrix-exponential action converges and is capped at dimension 10.
  • The bond dimension is exactly preserved without growth, making 1TDVP fixed-rank.The local exponentiation is the dominant cost.
  • O(d·N·D3·k) is the per-Trotter-step wall-clock scaling for k Krylov iterations.Per-site Krylov exponentiation acts on an O(ND2)-dimensional space.
  • On the Gaussian mixture target, the required bond dimension stays below the cap, so the fixed-D flow does not lose accuracy.

4.7 2TDVP (two-site)

2TDVP evolves and then SVD-splits two-site tensors, allowing adaptive bond-dimension growth up to Dmax while truncating singular values. This captures entanglement that 1TDVP cannot, at a per-step cost ∼ND higher, and is essential when target structure requires greater rank than the initial state.

  • 4.7 2TDVP (two-site): 2TDVP evolves a joint two-site tensor through Krylov exponentiation and SVD-splits it back into single-site cores.The split is Aj,j+1(t + ∆t) = UΣV †, followed by Aj ← U and Aj+1 ← ΣV †.
  • 4.7 2TDVP (two-site): tol = 10^-3 truncates singular values beyond the retained bond-dimension cap Dmax, which grows adaptively to capture entanglement structure unavailable to 1TDVP.The truncation condition is σD+1 < tol.
  • 4.7 2TDVP (two-site): ∼ND is the 2TDVP cost-per-Trotter-step factor relative to 1TDVP, driven chiefly by the larger Krylov subspace and the SVD.These are the principal additional computational terms.
  • Regime of advantage over 1TDVP: 2TDVP is essential when the equilibrium bond dimension exceeds the initial-state rank, such as for a correlated Gaussian mixture with genuine cross-axis covariance.In this regime, 1TDVP would remain locked at the initial bond dimension.

4.8 Subspace-expansion 1TDVP (1TDVP+SE) … 4.11 Pre-trained velocity-potential oracle and the trained MPS V -step

The paper develops subspace-expanded 1TDVP and pipeline-native, trained-MPS velocity potentials to balance tensor-network evolution cost, accuracy, and operator-construction overhead. It benchmarks these choices across analytic, learned, and MPS-based velocity oracles.

  • 4.8 Subspace-expansion 1TDVP (1TDVP+SE): 1TDVP+SE tracks dynamical bond demand by adding leading would-be-new singular values, interpolating between rank-preserving 1TDVP and bond-growing two-site 2TDVP.Pure 2TDVP scales as O(χ6), whereas 1TDVP scales as O(χ3) but stalls because it cannot represent perpendicular dynamical components.
  • 4.8 Subspace-expansion 1TDVP (1TDVP+SE): ∼2–7× wall-clock gap separates pure 2TDVP and 1TDVP+SE, with 2TDVP achieving lower final deviation while 1TDVP+SE is cheaper per step.For d=8, N=32, and K ∈ {40, 80, 120, 160}, 2TDVP self-limits at χ⋆≈16 through truncation.
  • 4.9 Targets and oracles: Gaussian mixtures use Tweedie’s identity for an analytic optimal velocity, while Swiss roll uses a trained JAM scalar-potential MLP oracle supplied to both classical and wavefunction-flow pipelines.The analytic velocity isolates tensor-network compression error from velocity-learning error.
  • 4.10 Pipeline-native training of Vt: JAM-trained Vt is pipeline-correct only at the dense N^d-grid fidelity used by its loss, with sufficiently fine integration substeps to make Trotter error negligible.Fixed MPS bond caps or coarser time discretization change the deployed pipeline’s sample-quality objective.
  • 4.11 Pre-trained velocity-potential oracle and the trained MPS V -step: A real-valued MPS velocity potential makes eiβVt diagonal-unitary and lets the V-step act directly from its cores, avoiding TCI compression into an MPO.This removes operator-construction work that dominates the runtime-TCI route; 2TDVP drives the trained-core scaling runs and bond-budget ramp.
  • 4.11 Pre-trained velocity-potential oracle and the trained MPS V -step: The trained MPS V-step eliminates per-substep operator construction for both runtime-TCI and build-then-apply routes, and all trained-MPS wavefunction-flow results use it.The operator eiβVt is never formed as an MPO.
  • 4.11 Pre-trained velocity-potential oracle and the trained MPS V -step: Nt=16 time-site encoding parameterizes Vt over continuous t through multilinear interpolation, while inference contracts the time site into spatial cores and aligns the oracle frame with the world grid.The spatial sites have local dimension N, and the source is shifted to the box center L/2 before the V-step.

4.12 Action-matching (JAM) training

JAM learns time-dependent potentials for the Swiss roll and Gaussian-mixture experiments using a sinusoidal-encoded MLP. The trained potential serves both as a classical gradient-flow baseline and as the velocity-potential oracle for wavefunction-flow evolution, isolating transport-method differences.

  • 4.12 Action-matching (JAM) training: JAM learns the potential Vt for the Swiss roll target and provides the MLP baseline for trained MPS potential ansätze on the Gaussian mixture.
  • 4.12 Action-matching (JAM) training: The MLP uses a 2d-dimensional periodic position encoding, a 32-dimensional sinusoidal time embedding, 3 hidden layers of width 128, SiLU activations, and a scalar output.The periodic encoding matches the wavefunction grid boundary, and ∇Vt is computed with PyTorch autograd.
  • 4.12 Action-matching (JAM) training: Training uses 50000 empirical samples, Adam at learning rate 10^-3, batch size 256, 2×10^4 iterations, no learning-rate schedule, and a held-out 10000-sample validation split.Samples are formed by interpolating between x0 and x1 after drawing t uniformly from [0, 1].
  • 4.12 Action-matching (JAM) training: The trained Vt is evaluated on the position grid as both the JAM classical gradient flow and the wavefunction-flow velocity-potential oracle.Because both pipelines use the same Vt, they differ only in how they transport the velocity-potential oracle.

4.13 Hyperparameter search protocol · 4.14 Bond-dimension ramp schedule · 4.15 Sample quality evaluation metrics

The study uses per-dimension Sobol hyperparameter searches and a stepwise bond-dimension ramp for higher-dimensional MPS evolution. Sample quality is assessed through direct conditional sampling, sliced Wasserstein distance, MMD, and a finite-sample target–target reference.

  • 4.13 Hyperparameter search protocol: Sobol-sample (N, K, Dmax) over a dimension-specific box, recording SW, MMD, χmax, and total wall-clock time for every cell.Each cell runs as a Modal cloud function on a single-CPU container with 16–64 GB memory and incremental checkpointing.
  • 4.13 Hyperparameter search protocol: At d ≤4, searches reach N = 128 or N = 256, whereas d ≥5 is limited to N ≤32 by container memory.Each cell uses 1–3 seeds.
  • 4.13 Hyperparameter search protocol: Cells whose N d > 109 would exceed dense-grid memory, so those cells run only the MPS methods.The search-grid N cap is set by container memory for the dense-grid path.
  • 4.14 Bond-dimension ramp schedule: Higher-dimensional trained MPS potential experiments can replace fixed Dmax with a stepwise Dmax(k) schedule that grows with Trotter-step index k.With a Gaussian source state having χ = 1, setting Dmax = 64 from k = 0 forces repeated truncation before sufficient entanglement develops.
  • 4.14 Bond-dimension ramp schedule: Schedules are sorted (kthresh, Dmax) pairs, and the active cap is the Dmax associated with the largest threshold satisfying kthresh ≤k.For Table 2 scaling runs, the cap starts at χ0 = 1 or χ0 = 3 and reaches its final value over roughly the first 112 Trotter steps.
  • 4.15 Sample quality evaluation metrics: Samples are obtained by direct conditional (perfect) sampling from the evolved wavefunction MPS, site by site, without a Markov chain.Quality uses sliced Wasserstein distance with 128 random projections and radial-basis-function MMD.
  • 4.15 Sample quality evaluation metrics: MMD uses an isotropic Gaussian kernel following standard implementations.The reference scale is finite-sample dependent rather than an irreducible error floor.
  • 4.15 Sample quality evaluation metrics: The target–target Exact reference averages SW over 32 pairs of independent n = 500 target draws, using independent random projections per pair.At n = 500 and L = 128, the Gaussian-mixture reference ranges from 0.069 at d = 2 to 0.060 at d = .

4.16 Rare-event sampling · 4.17 Use of large language models

Rare-event sampling uses an analytically defined tail, cost-optimized amplitude amplification, and exact Grover statistics to validate asymptotic scaling while accounting for finite-probability behavior. The authors also used language models and coding assistants under author direction, with review and testing of AI-assisted code.

  • 4.16 Rare-event sampling: Rare events are defined with the nearest-mode k-sigma rule and analytic tail probability p_rare(k) = Pr[χ2Born samples closer to the domain center than any mode are excluded before forming the tail fraction.
  • 4.16 Rare-event sampling: Amplitude amplification selects Grover power n by minimizing expected cost (2n+1)/P(n), rather than maximizing single-shot success probability.Here, the textbook single-shot rule costs 11% more on average and produces a stepped cost curve.
  • 4.16 Rare-event sampling: The 1/√p_rare law is asymptotic: optimal round count reaches zero beyond that regime, costs flatten at the one-preparation floor, and curves meet.This describes the finite-probability departure from the asymptotic scaling.
  • 4.16 Rare-event sampling: The study simulates exact Grover outcome statistics, using p_rare from the prepared state’s Born-sample tail fraction.The flow ODE pipeline uses the same learned potential and 640 substeps, making oracle error common to both pipelines.
  • 4.17 Use of large language models: Claude Opus 4.7, 4.8, and 5 and Claude Code were used to improve language and readability and assist research-software development.These tools supported writing and coding rather than study design, theory derivation, or result interpretation.
  • 4.17 Use of large language models: All AI-assisted code was written under the authors’ direction and reviewed and tested before use in reported experiments.The passage states that large language models were not used to design the study, derive the theory, or interpret results.

Competing interests

All authors are employees of Sygaldry Technologies, Inc., and some hold equity or equity-based compensation; C.R. and M.J.K. are co-founders.

  • All authors are employees of Sygaldry Technologies, Inc., which develops algorithms and hardware for quantum-accelerated artificial intelligence; some hold equity or equity-based compensation, and C.R. and M.J.K. are co-founders.

S1 Theory

Wavefunction flows construct conservative transport as unitary evolution, so the final Born distribution approximates the target while enabling amplitude-estimation speedups for expectation values. Tensor-network structure supplies the resource bounds and compression rationale, with discretization and product-formula errors limiting digital accuracy.

  • Wavefunction-flow construction: A stochastic interpolant defines a continuity-equation flow from p0 to p1, with conservative velocity vt = ∇Vt enabling a quantum lift.The interpolant transports samples along the ODE ˙xt = vt(xt), and the potential can be characterized variationally.
  • Wavefunction-flow construction: The Hamiltonian Hc_t = i[K, Vt] decouples velocity from wavefunction phase, avoids a quantum-potential correction, and makes the V-step position-diagonal.Unlike canonical kinetic-plus-potential evolution, this construction is not a natural Hamiltonian but is amenable to MPS compression.
  • Amplitude estimation: O(ε^-1) queries to Uprep let amplitude estimation estimate bounded expectations from the final qsample, providing the construction’s quadratic sampling motivation.The observable is embedded as a position-diagonal operator, and Uprep prepares the evolved wavefunction from a fiducial state.
  • Product formula: O(Δt^2) local error follows from the symmetric eight-letter group-commutator formula, whose degree-three corrections cancel while K and Vt split locally.K is separable across coordinates and exponentiates through one-dimensional FFTs, whereas Vt is diagonal in position.
  • Tensor-network resources: The rank-2d bound applies to the mixture density but not √pt, Vt, its phase, or the propagated wavefunction, so adequate rank caps remain empirical.Observed propagated-MPS bond dimensions stabilize compatibly with D ∈ {16, 32} for d ≤ 8.
  • Tensor-network resources: O(N^d log N) dense-grid time and O(N^d) memory contrast with MPS V-step costs O(dND^3) for 1TDVP and O(dND^2m) for TCI.MPS memory is O(dND^2); assuming D scales linearly with d, 1TDVP wall-clock time scales as d^4.

S1.8 The bond dimension of Vt as the unified resource parameter

The bond dimension D_V is presented as a unified structural resource governing classical TCI queries, compiled quantum-gate costs, and TDVP wall-clock time and memory. The resulting polynomial-resource claims require additional compressibility assumptions and have explicit precision, initialization, and state-loading caveats.

  • Unified resource parameter: D_V simultaneously governs TCI oracle queries, precompiled V-step gate counts, and empirically measured TDVP wall-clock time and memory.The section identifies bond dimension as a shared structural parameter across these classical and quantum costs.
  • Structural assumption: Smoothness controls spectral discretization error but does not itself imply small D_V; compressibility is an additional assumption depending on target geometry.The text links bond dimension to the number and spatial arrangement of modes and verifies compressibility analytically or empirically in specific targets.
  • Per-step construction: TCI exactly reconstructs a rank-D_V MPS when every unfolding has rank at most D_V, while approximate compression requires η_V/|β| generator accuracy for per-step operator tolerance η_V.The refinement-sweep count depends on unfolding singular values, and no general sweep-count guarantee is available.
  • Total resource cost: Both total classical query and quantum gate costs are polynomial in d whenever D_V is polynomial in d.The total-cost corollary assumes uniform bond-dimension control over t and uses η_V = O(ϵ/r) for r Trotter steps.
  • Gaussian-mixture specialization: For orthogonal-mode Gaussian mixtures, assuming D_V = O(d) on [τ, T] yields degree 9 in explicit d factors before substituting the grid bound, with exponentially small cost-to-dense-grid ratio.The assumption is empirical and consistent with the reported runs; the dense-grid reference cost is Θ(N^d).
  • Limitations: The analysis starts at t = 10^-3 because the analytic potential is singular as t → 0, and final-state loading remains a separate primitive from generator costs.Fixed-point rounding in the compiled V-step also lacks a performed error analysis.

S2 Supplementary results

Supplementary results identify TCI+1TDVP as the robust Pareto-optimal method, while showing that flow difficulty increases for sharper modes and finer grids. They also establish cost–accuracy tradeoffs and confirm that a trained MPS potential reproduces analytic-potential behavior with small velocity-learning error.

  • Pareto performance: TCI+1TDVP is robust and Pareto-optimal, with SW=0.14–0.21 across d=2–5 at small D* and interior K*.At d∈{6,7,8} with D=64, 1TDVP remains accurate and tractable, completing in tens of seconds, while runtime-TCI TCI+2TDVP exhausts 32 GB.
  • Sharpness and resolution: At σ=0.35 (σ/dx=1.4), instantaneous deviation reaches up to ∼0.15 near t=1, whereas σ=1.0 (σ/dx=4.0) tracks the reference almost exactly.Across all widths, 2TDVP stays closer to the probability-flow reference than 1TDVP+SE at matched K; increasing K helps mainly for sharp modes.
  • Sharpness and resolution: Along the resolution route, deviation grows with N, opposite to the sharpness route at matched σ/dx, because grid refinement also raises the Trotter constant.The Trotter constant scales as d2N 4, so refining the grid does not by itself ease the flow and demands proportionally more effort.
  • Cost–accuracy tradeoff: 2TDVP reaches lower deviation than 1TDVP+SE but at roughly an order of magnitude more wall-clock time.The cost–accuracy comparison uses total deviation against measured wall-clock time across Trotter count K, mode width σ, and V-step choice.
  • Velocity-source validation: The trained MPS potential reproduces the analytic-potential flow, with controlled small deviation that shrinks with K and isolates velocity-learning error as small.This comparison uses σ=0.5, d=8, and N=32 across instantaneous and accumulated deviations.
Loading 2608.21700v1…