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Quantum Frozen--Oseen homotopy analysis method with LCHS for solving nonlinear partial differential equations

Jinhao Yuan, Leyu Chen, Tiegang Liu, Kun Wang

arXiv:2609.39170v1math.NAcs.NAquant-ph

TL;DR

Fine-grid nonlinear PDE simulations are computationally demanding, and quantum linear-evolution methods do not directly propagate nonlinear coupling. FOQHAM freezes the full Fréchet derivative at a prescribed profile and product-lifts finite homotopy hierarchies for LCHS propagation. Classical tests support accurate non-iterative approximations in the tested regimes, with reported limitations on access and signal assumptions.

  • Problem

    Fine grids, stiffness, and long integration intervals make nonlinear PDE simulation computationally and storage intensive, while quantum linear-evolution primitives do not directly propagate nonlinear coupling.

    Source: §1 · p. 2

  • Method

    FOQHAM freezes the full Fréchet derivative at a prescribed profile and product-lifts each finite-order homotopy hierarchy into a fixed affine linear system propagated with LCHS.

    Source: §1 · p. 3, §5 · p. 46

  • Results

    Classical tests support accurate non-iterative approximations across the tested PDEs; matched controls link reduced homotopy order to a smaller lifted representation in a Burgers case, while the KdV comparison favors mean transport at the prescribed accuracy.

    Source: §1 · p. 3, §5 · p. 46

  • Takeaways & Limitations

    The framework connects nonlinear PDE approximation to quantum linear evolution, with bounds characterizing conditions under which auxiliary choice can improve approximation.

    Source: §5 · p. 46

    The reported finite-rule field errors do not verify Theorem 4’s access and signal assumptions, and normalization can increase substantially in the cited Navier–Stokes run.

    Source: §3.4 · p. 28

Abstract

from arXiv · show

Nonlinear partial differential equations (PDEs) underpin computational fluid dynamics, yet resolving nonlinear transport on fine grids remains computationally demanding. Quantum linear-evolution algorithms offer a possible route to large-scale simulation but cannot directly propagate nonlinear coupling. Here we develop a Frozen--Oseen quantum homotopy method (FOQHAM) framework that links transport-aware auxiliary-operator selection to the size of the resulting linear representation. We freeze the full Fr\'echet derivative at a prescribed flow profile, retaining transport and profile-gradient coupling and close each prescribed finite-order homotopy hierarchy as a fixed affine linear system by product lifting. We formulate its propagation through Linear Combinations of Hamiltonian Simulations (LCHS), without outer homotopy iterations or profile updates. For spatially semidiscrete equations, we establish local approximation bounds and exact closure of the finite-order hierarchy. Classical tests on Burgers, Korteweg--de Vries, Zakharov--Kuznetsov, and one- and two-dimensional isothermal compressible Navier--Stokes equations provide numerical evidence of accurate non-iterative approximations in the tested regimes. The framework thus connects nonlinear PDE approximation to quantum linear evolution and offers a conditional path toward quantum acceleration.

1. Introduction

Fine-grid nonlinear PDE simulations are computationally demanding, while quantum linear-evolution primitives do not directly propagate nonlinear coupling. FOQHAM freezes transport-aware dynamics and product-lifts finite homotopy hierarchies; classical tests support accurate non-iterative approximations in the tested regimes.

1. Introduction

  • Quantum linear-system and evolution methods address linear dynamics, but nonlinear transport couples unknown components through products they cannot directly propagate.Carleman methods use an infinite lifted hierarchy with problem-dependent truncation bounds, while QHAM closes a prescribed finite hierarchy by lifting coefficient products.Source: §1 · p. 2
  • FOQHAM freezes the full Fréchet derivative at a prescribed profile, retaining transport and profile-gradient coupling alongside the residual and nonlinear remainder.The profile and lifted generator remain fixed during propagation.Source: §1 · pp. 2–3
  • Product lifting closes each prescribed finite-order homotopy hierarchy as a fixed affine linear system that LCHS propagates without outer homotopy iterations or profile updates.For spatially semidiscrete polynomial systems, the framework gives local approximation bounds and exact finite-order closure.Source: §1 · p. 3
  • Tests cover Burgers, Korteweg–de Vries, Zakharov–Kuznetsov, and one- and two-dimensional isothermal compressible Navier–Stokes, showing that more retained linearized physics can lower homotopy order without minimizing representation size.The introduction also notes that separate coefficient controls isolate auxiliary selection and that resource analysis retains matrix-access, normalization, source-integration, and readout costs.Source: §1 · p. 3

2.1. Frozen–Oseen homotopy analysis

The section sets up a spatially semidiscrete nonlinear problem and decomposes it around a fixed profile using the frozen Fréchet derivative. It then gives local bounds, sufficient convergence conditions, and profile-selection guidance for finite homotopy approximations.

2.1.1. Problem setting and boundary treatment

  • The problem is an autonomous finite-dimensional system after spatial semidiscretization; time-independent nonhomogeneous boundary data are removed by a lifting whose known terms enter the forcing and operator.The norm estimates concern the method-of-lines system, and local bounds may depend on the active dimension.Source: §2.1.1 · pp. 3–4

2.1.2. Exact frozen–Fréchet decomposition

  • Freezing the Fréchet derivative at a fixed profile gives an exact regrouping of the dynamics that retains the nonlinear remainder in the homotopy hierarchy.The state is written as the prescribed profile plus a correction, and the frozen operator is the Jacobian at that profile.Source: §2.1.2 · p. 4

2.1.3. Second-order remainder bound

  • The frozen–Oseen remainder is O(∥w∥^2), improving locally on the O(∥w∥) correction from retaining only the original linear part, when the frozen semigroup is sufficiently controlled.The quadratic bound applies when the segment from the profile to its perturbed state remains in the specified neighborhood.Source: §2.1.3 · p. 5

2.1.4. Mild solution and contraction criterion

  • Under the stated local second-derivative and frozen-semigroup bounds, a sufficient contraction condition yields a unique mild solution in the specified radius-R ball.The contraction acts on the Duhamel map over the finite time interval.Source: §2.1.4 · p. 6

2.1.5. Conditional comparison with the original linear operator

  • A smaller contraction upper bound is conditional: the reduced nonlinear Lipschitz constant must outweigh any increase in transient amplification, with the self-map condition holding for both decompositions.The comparison is between the frozen decomposition and one using the original linear operator.Source: §2.1.5 · pp. 6–7

2.1.6. Homotopy deformation hierarchy

  • Analyticity and contraction conditions give a sufficient homotopy-tail bound, while convergence of the semidiscrete deformation series identifies its limit with the unique solution in the ball.Product lifting closes each prescribed polynomial hierarchy; certified orders compare representation sizes among certified choices, not the actual required order or total quantum cost.Source: §2.1.6 · pp. 8, 10

2.1.7. Frozen-profile construction and selection

  • Profiles are fixed before propagation; changing the profile affects the finite-order tail, not the physical model, and the tests do not rank profiles across PDEs.Initial-state freezing removes the initial correction; Burgers layer enrichment and the Navier–Stokes acoustic mean target different transport regimes. The perturbation bound also gives a robustness margin when the candidate’s logarithmic norm is negative.Source: §2.1.7 · pp. 10–12

2.2. Quantum-compatible linearization by product lifting

Product lifting closes each prescribed finite-order homotopy hierarchy as an affine linear system, with exact closure and target recovery. The lifted representation grows with homotopy order, while exact symmetry reduction and sparse-access conditions affect its implementation.

2.2.1. Finite closure of the quadratic hierarchy

  • The quadratic product-lifted hierarchy closes exactly, so the remaining approximation is the homotopy tail rather than an additional truncation of lifted variables.Differentiation preserves the defined weight, and residual insertion lowers it.Source: §2.2 · p. 12, §2.2.1 · p. 12

2.2.2. Lifted dimension and target recovery

  • The target block recovers the physical approximation, and exact symmetry reduction can compress the lifted representation only when initialization, dynamics, and target recovery are preserved.Discarding required product variables introduces an additional approximation.Source: §2.2.2 · pp. 12–13

2.2.3. Extension to general polynomial nonlinearities

  • For polynomial degree d, a finite set of weighted product variables is closed under product-rule dynamics, yielding a fixed affine system without nonlinear products during propagation.The maximum tensor degree is bounded by the theorem’s index set; consistent initialization recovers the truncated deformation hierarchy.Source: §2.2.3 · pp. 13–14
  • An exact product-symmetry quotient preserves trajectories and target fields, but a finite LCHS rule on the scaled compressed generator need not match projection of the same rule on the ordered generator.The stated equivalence is for exact affine dynamics; similarity scaling does not preserve norms in the scaled coordinates.Source: §2.2.3 · p. 15

2.2.4. Sparse-access requirements

  • Sparse-Hamiltonian implementation requires efficient row and column access, while dense residual insertion can destroy column sparsity and requires its preparation cost and normalization to be accounted for.The local ordered Navier–Stokes construction does not certify the Fourier or symmetry-compressed production layouts.Source: §2.2.4 · p. 16

2.3. LCHS realization of the lifted linear dynamics

LCHS represents the fixed lifted affine dynamics through finite weighted unitary evolutions, with separate controls for homogeneous and source propagation. Scaling, spectral shifting, and finite-rule errors impose additional accuracy and normalization considerations, while the workflow keeps the profile and lifted matrix fixed.

2.3.1. Homogeneous and source propagation

  • The homogeneous and source propagators are approximated by finite LCHS rules, with the experiments using an equally spaced finite sum for the k-integral.The cited bound for the raw finite-sum operator error applies for L ⪰ 0 and sufficiently small h.Source: §2.3.1 · pp. 17–18

2.3.2. Separate K1 and K2

  • Separate cutoffs and node counts control homogeneous and source propagation, and source accuracy requires a uniform propagator bound plus joint cutoff and node refinement.Interval refinement alone can approach the wrong generator.Source: §2.3.2 · p. 18

2.3.3. Similarity scaling and spectral shifting

  • Similarity scaling preserves the exact physical solution and eigenvalues, but balancing the matrix can worsen state preparation; spectral shifting adds a normalization factor.The initial target probability does not bound the final one, and an operator-level guarantee requires a certified lower spectral bound.Source: §2.3.3 · pp. 18–19

2.3.4. Error budget

  • Finite-rule errors depend on the interval-start state, source norm, and subsequent propagation, so the experiments report physical-field, spatial, and complete-lift errors separately.Source: §2.3.4 · p. 19

2.3.5. Non-iterative workflow

  • The workflow freezes the profile and lifted matrix throughout; its time loop composes linear propagation rather than iterating HAM updates.Source: §2.3.5 · p. 19

3. Resource analysis and conditional scaling comparison

The resource analysis distinguishes explicit lifted-state costs from a conditional quantum bound for sparse-observable outputs. The bound depends on access, signal, and uniformity assumptions, and the reported finite-grid errors do not verify those assumptions or establish a matched quantum advantage.

3.1. Explicit classical lifted evolution

  • Explicit lifted evolution incurs costs for storing and evolving the full lifted state, while writing that state takes Ω(Dm) time and few-observable tasks have different classical baselines.Direct nonlinear PDE solvers and low-rank or symmetric-tensor implementations can avoid materializing all Dm coordinates.Source: §3.1 · p. 20

3.2. Quantum access model

  • Quantum resource estimates depend on access and output requirements, and large normalization-to-signal ratios in the examples leave the theorem’s non-small-signal premise unverified despite small classical field errors.The reported ratios are diagnostics, not measured circuit success probabilities or lower bounds for other implementations.Source: §3.2 · pp. 20, 23, 24

3.3. Conditional quantum upper bound and scaling comparison

  • The conditional quantum upper bound requires uniform homotopy-tail, coherent-access, signal, and error-control assumptions; it supplies quantum-side conditions, not a matched classical query lower bound.Assessing same-observable advantage also requires accounting for direct classical PDE solvers, profile generation, loading, source-time quadrature, and output preparation.Source: §3.3 · pp. 26–27

3.4. Dependence on spatial resolution

  • For standard finite differences, derivative norms can grow with grid resolution, and the reported Navier–Stokes shift gives e^(δshT) ≃ 6.1 × 10^15, a normalization factor rather than a measured failure probability.Theorem 4’s access and signal assumptions are not verified by the finite-rule field errors.Source: §3.4 · p. 28

4. Numerical experiments

The section tests a fixed-profile Frozen–Oseen hierarchy on a semidiscrete Burgers benchmark, then compares auxiliary choices and profile-based approximations against same-grid references.

4.1.3. Discretization and results

  • The comparison fixes the physical target, grid, initial data, and output norm; conventional HAM and IQHAM are classically integrated, while Frozen–Oseen uses fixed linear substeps and symmetric-product compression.The other two hierarchy curves are not LCHS results.Source: §4 · p. 28, §4.1.3 · p. 31

4.1.1. Governing equation and exact Oseen split

  • The Burgers benchmark adds a stationary body force, uses homogeneous Dirichlet boundaries, and freezes the full Fréchet–Oseen operator at a prescribed profile.The construction retains the residual and exact nonlinear remainder.Source: §4.1.1 · pp. 28–29

4.1.2. Frozen profile

  • The fitted profile uses an O(ν)-width boundary-layer enrichment, satisfies both boundaries, and is fitted once for the benchmark, making the main-case errors in-sample.The layer balance motivates the width and tanh factor, but not the fitted constants.Source: §4.1.2 · p. 30

4.1.3. Discretization and results

  • The analytic profile derivatives define an enriched semidiscrete model, but the n = 5 grid does not resolve its O(ν) layer; the same-grid reference differs from a 400-node PDE proxy by 8.51 × 10−3.Changing the enrichment would change the spatial model.Source: §4.1.3 · pp. 30–31
  • Four unfitted n = 17 holdouts improve at order two but remain grid-underresolved, with n = 17 to 129 discrepancies of 5.23 × 10−2–8.53 × 10−2.The holdouts use the initial condition as the prescribed profile.Source: §4.1.3 · p. 31
  • Transport improves all four fifth-order controls over diffusion, but the full Jacobian is not uniformly best; in the first case it reaches 1% error one order before frozen advection, reducing dimension by factors 7.46, 12.94, and 23.71 across three grids.These are classical hierarchy representation counts, not measured quantum speedups.Source: Table 3 · p. 32, §4.1.3 · pp. 32–33

4.2. Korteweg–de Vries equation

The KdV benchmark uses a fixed initial-wave profile on a seven-point periodic Fourier grid to compare Frozen–Oseen auxiliary choices, discretization error, and finite-LCHS propagation. Results show that auxiliary selection affects the homotopy order and representation size, while finite-LCHS propagation discrepancies remain distinct from spatial error.

4.2.1. Periodic traveling-wave benchmark

  • For this cnoidal-wave family, the Frozen–Oseen remainder is O(η^2), whereas a dispersion-only auxiliary leaves first-order transport in the forcing.This local comparison does not establish uniform convergence.Source: §4.2.1 · pp. 33–34

4.2.2. Discretization and error control

  • Spatial error falls from 1.5663 × 10^-5 on seven points to 6.3359 × 10^-8 on nine and 3.9201 × 10^-12 on 13 for this smooth wave.These are nonlinear spatial errors, distinct from propagation error.Source: §4.2.2 · p. 34
  • Order four reaches the spatial-error level, and all LCHS/direct wave discrepancies are below 6.2 × 10^-9.The hierarchy uses exact symmetric-product compression and four fixed LCHS intervals.Source: §4.2.2 · p. 34

4.2.3. Matched auxiliary controls and HAM recipes

  • Mean transport reaches the 10^-4 PDE-wave target at order two, one order before full G_n; both auxiliaries approach the spatial-error level by order five.The comparison fixes ℏ = −1 and zeroth-coefficient initialization to isolate auxiliary choice.Source: §4.2.3 · p. 35
  • In the separate recipe comparison, order-five dispersion-auxiliary HAM has wave error 0.66028, while IQHAM reaches 1.5626 × 10^-5 after nine outer updates.Those recipes differ in zeroth trajectory and ℏ, unlike the matched controls.Source: §4.2.3 · p. 35, Figure 2 · p. 36
  • At a common 10^-4 target, mean transport uses a smaller complete lift and lower output-aware query proxy than full G_n.The proxy is not a gate count or measured runtime.Source: §4.2.3 · p. 36, Table 6 · p. 37

4.3. Two-dimensional Burgers and Zakharov–Kuznetsov equations

The two-dimensional tests apply first-order Frozen–Oseen HAM-LCHS to Burgers and Zakharov–Kuznetsov (ZK) fields with profiles frozen at their initial states. Both tests report field errors and complete-lift propagation discrepancies against their respective references.

4.3. Two-dimensional Burgers and Zakharov–Kuznetsov equations

  • Both initial profiles are frozen, and the tests use 9 × 9 Fourier grids, first-order lifts, four fixed linear substeps, and tensor scale 0.1.Burgers is tested at T = 1 and ZK at T = 0.5.Source: §4.3 · p. 37
  • For Burgers and ZK, first-order Efluc measures 3.49 × 10−5 and 7.03 × 10−4, respectively; their corresponding complete-lift LCHS/direct discrepancies are 3.06 × 10−9 and 4.62 × 10−9, respectively.Source: §4.3 · p. 37
  • Figure 3 presents Burgers u and v and ZK u as reference, first-order result, and pointwise signed error fields.Reference/result pairs share a scale, while error maps use separate zero-centered scales; 128 × 128 resampling is for display only.Source: Figure 3 · p. 38

4.4. One-dimensional isothermal compressible Navier–Stokes equations

The one-dimensional isothermal compressible Navier–Stokes tests compare constant kinematic and dynamic viscosity using profiles derived from a linear acoustic–viscous predictor. The reported field errors decrease with homotopy order in the tested cases, with an endpoint-profile trade-off for dynamic viscosity.

4.4. One-dimensional isothermal compressible Navier–Stokes equations

  • The tests distinguish constant kinematic viscosity η(ρ) = νρ from constant dynamic viscosity η(ρ) = µ, with ν = µ = 0.005 and T = 1.5.The initial conditions are ρ(x, 0) = e^(0.3 cos x) and u(x, 0) = −0.6 sin x on a periodic interval.Source: §4.4 · p. 39
  • The fixed profile comes from a linear acoustic–viscous predictor; its time average is the constant profile closest in integrated squared distance to that predictor and uses no nonlinear reference.For constant µ, an endpoint-average profile is also tested, and lifted initial products account for the profile differing from the initial state.Source: §4.4 · p. 39
  • With the acoustic-mean profile, E_fluc falls from 1.04 × 10^-2 to 5.81 × 10^-4 by order three for constant ν, and to 2.373 × 10^-3 by order two for constant µ.For µ, the endpoint-average profile improves pooled first-order error to 8.85 × 10^-3 but increases velocity-component error from 1.260% to 1.396%.Source: §4.4 · p. 39
  • The constant-ν and constant-µ tests use 17 and 13 Fourier points, respectively, with symmetry reduction and twelve fixed linear substeps.Density is reconstructed classically as ρ = e^q after interpolation to the 129-point reference grid.Source: §4.4 · p. 39, Table 7 · p. 40

4.5. Two-dimensional isothermal compressible Navier–Stokes equations

The two-dimensional compressible tests compare first-order Frozen–Oseen approximations with refined nonlinear references, then extend the evaluation to an exploratory incompressible vortex pair and matched classical propagation. The results separate approximation error from propagation error and do not establish a quantum speedup.

4.5. Two-dimensional isothermal compressible Navier–Stokes equations

  • Across all six compressible runs, complete-lift LCHS/direct discrepancies stay below 4.4 × 10−8, distinct from the remaining approximation error.In refined compression, the u-component error is still above 1%.Source: §4.5 · p. 41
  • At 11 × 11, Efluc is 1.632 × 10−3 for coupled compression and 1.077 × 10−3 for vortex–acoustic interaction; refined-compression u error remains 1.13 × 10−2.The comparison uses first-order finite-LCHS results with time-averaged frozen profiles against a 31 × 31 nonlinear reference; Figures 5 and 6 show fields and signed errors.Source: §4.5 · p. 41, Figure 5 · p. 42, Figure 6 · p. 43

4.6. Exploratory two-dimensional incompressible vortex-pair test

  • The exploratory vortex-pair test reports final vorticity error 7.33 × 10−3 against the 31 × 31 reference, with a 2010-dimensional first-order lift padded to 2048.This remains an exploratory first-order test; the 11 × 11 spatial discrepancy is 5.95 × 10−3 and complete-lift LCHS/direct discrepancy is 1.13 × 10−7.Source: §4.6 · p. 44, Figure 7 · p. 45

4.7. Matched linear-observable comparison

  • At the matched small-grid settings, DOP853 is more accurate and faster than explicit lifted propagation, and the comparison establishes neither quantum speedup nor a necessary classical storage baseline.The comparison uses Burgers n = 5 and constant-µ NS Nx = 13, with DOP853 relative tolerance 10−3; reported classical times are not quantum runtimes.Source: §4.7 · p. 46, Table 9 · p. 46

5. Conclusions and discussion

FOQHAM uses transport-aware auxiliary selection and product lifting to represent prescribed finite-order homotopy hierarchies as fixed affine systems for LCHS propagation. Tests support accurate non-iterative approximations in the tested regimes, while implementation and resource claims remain conditional.

5. Conclusions and discussion

  • Freezing the full Fréchet derivative retains transport and profile-gradient coupling, while product lifting closes each prescribed finite-order polynomial hierarchy as a fixed affine linear system.LCHS propagates this system without outer homotopy iterations or profile updates; local approximation and profile-sensitivity bounds characterize when the auxiliary can improve approximation.Source: §5 · p. 46
  • Tests on Burgers, Korteweg–de Vries, Zakharov–Kuznetsov, and isothermal compressible Navier–Stokes support accurate non-iterative approximations in the tested regimes.Matched auxiliary controls connect reduced homotopy order to a smaller lifted representation in one Burgers case, whereas the KdV comparison favors mean transport over the full frozen derivative at the prescribed accuracy.Source: §5 · p. 46
  • Auxiliary selection is a joint approximation-and-resource-design problem involving nonlinear remainder, propagator growth, and representation size.The resource analysis is a basis for quantum implementation only under stated matrix-access, normalization, signal, and readout conditions.Source: §5 · p. 47
  • Structured compressed encodings and independently validated profiles are identified as next steps toward larger, spatially resolved systems.Source: §5 · p. 47

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