Simulating Time Dependent and Nonlinear Classical Oscillators through Nonlinear Schr\"odingerization

Abhinav Muraleedharan; Nathan Wiebe

REVIEW 4 major objections 5 minor 1 cited by

Simulating Time Dependent and Nonlinear Classical Oscillators through Nonlinear Schr\"odingerization

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that forced, time-dependent, and weakly nonlinear classical oscillator systems can be reduced—through a method it calls Nonlinear Schrödingerization—to Hermitian quantum evolution, with query cost near-linear in time and…

desk verdict The forced-oscillator reduction is a real idea, but the symmetrization step is not Hermitian, so the nonlinear theorems don't follow. read the letter →

arxiv 2505.17170 v1 pith:T4MXNBJV submitted 2025-05-22 quant-ph

Abhinav Muraleedharan , Nathan Wiebe This is my paper

classification quant-ph

keywords quantumsimulationcoupledclassicaloscillatorsnonlinearSchrödingerequationCarlemanlinearizationtime-dependentforcessymmetrizedoperatorforcedoscillatorsystems

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 tries to show that a broad class of classical oscillator systems—networks of $2^n$ coupled masses with time-dependent external forces, time-varying stiffness matrices, and weak quadratic nonlinearities—can be simulated on a quantum computer with cost that grows almost linearly in evolution time and only logarithmically in the number of masses. The central move, called Nonlinear Schrödingerization, embeds the non-conservative or nonlinear dynamics into a larger conservative system, reduces it to a nonlinear Schrödinger equation, and then re-linearizes that equation in a higher-dimensional space through a new symmetrization step. If the construction holds, every system in this class inherits the exponential speedup previously shown for simple harmonic oscillator networks, and the main obstacle—lack of energy conservation—is removed. A sympathetic reader would care because this substantially widens the set of classical mechanical problems for which quantum simulation is provably efficient.

What carries the argument

The central object is the symmetrized Carleman operator $\hat{Q}(\eta)=\sum_i(|i\rangle\langle i|\otimes A_{i+1}+\frac{1}{\eta}|i\rangle\langle i+1|\otimes B_{i+1}+\frac{1}{\eta}|i+1\rangle\langle i|\otimes B^\dagger_{i+1})$, a Hermitian matrix built from the original linear and nonlinear coupling blocks $A_i$ and $B_i$. It carries the argument by turning the non-unitary truncated Carleman system into a norm-preserving Schrödinger evolution: the scaling $\eta$ suppresses the non-Hermitian back-action terms, and Lemma 32 gives a sufficient $\eta$ for the rescaled state $D|\hat{p}(t)\rangle$ to approximate the true Carleman state $|p(t)\rangle$ within $\epsilon$. Forced and time-dependent oscillator problems are fed into this machine by constructing higher-dimensional oscillator systems whose subspaces reproduce the desired non-conservative dynamics.

What would settle it

Take the concrete Carleman matrix $H$ from Eq. (175) with $B_i$ blocks not paired with adjoints, compute $\|e^{-iHt}\|_2$ for small $t$; if it exceeds $1+\delta$ for arbitrarily small $\delta$, or if a direct numerical integration of the symmetrized system fails to track the truncated Carleman system within the claimed $\epsilon$, then Lemma 32's bound is false and the theorems relying on it lose their support.

Watch

Extended reading notes

Core claim

The paper's central claim is that the dynamics $M\ddot{x}=-Kx+f(t)$ (forced), $M\ddot{x}=-K_1x+K_2x\otimes x$ (nonlinear), $M\ddot{x}=-K(t)x$ (time-dependent stiffness), and the combined forced time-dependent case can each be mapped to Hermitian time evolution. For the forced case, auxiliary masses with carefully chosen initial conditions enact the external force while their large mass $m_f$ suppresses back-action; perturbation theory bounds the error by $\epsilon$ once $m_f \ge (t\|K'\|\cdots)(1+\Xi(0)/\epsilon)$. For nonlinear and time-dependent cases, Carleman linearization embeds the nonlinear Schrödinger equation $|\dot{\psi}\rangle=-iH_1|\psi\rangle+H_2|\psi\rangle\otimes|\psi\rangle$ into a high-dimensional linear system, and a scaling parameter $\eta$ symmetrizes the non-Hermitian Carleman generator so that Hamiltonian simulation applies. The stated complexities are $O(\tau+\log(1/\epsilon))$ for forced oscillators with $\tau=t\sqrt{2\alpha d}$, and $O(\alpha k^2 t + k\log(\cdots))$ for nonlinear Schrödinger equations, with the Carleman truncation order $k$ logarithmic under non-resonance conditions.

Load-bearing premise

The construction depends on the assumption that the unperturbed linearized system evolves by a norm-preserving unitary; in the symmetrization lemma this is applied to a matrix that is not Hermitian, because the lower off-diagonal blocks are not the adjoints of the upper ones, and that gap is what all later error bounds inherit.

Editorial extensions

If this is right

For forced oscillator systems, the algorithm takes $O(\tau+\log(1/\epsilon))$ oracle queries, with $\tau=t\sqrt{2\alpha d}$, and uses $O(\log(N(l+1)))$ qubits.
Weakly nonlinear Schrödinger equations with non-increasing norm and no resonances are simulable with $O(\alpha k^2 t + k\log(\cdots))$ queries, and the Carleman truncation order is logarithmic in $T/\epsilon$ when the resonance gap $\Delta$ is large enough.
Nonlinear oscillator networks with quadratic nonlinearities, bounded state norm, and non-resonant spectra reduce to the nonlinear Schrödinger simulation and inherit the same complexity.
Time-dependent stiffness matrices, and time-dependent stiffness combined with external forces, are embedded in higher-dimensional nonlinear oscillator systems and then simulated with the same machinery.
If correct, all these classes inherit the exponential separation from classical simulation previously established for the conservative linear oscillator case.

Reading between the lines

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

Editorial inference: the same perturbative embedding of non-conservative terms into larger conservative systems should apply to other linear non-unitary dynamics, such as damped oscillator networks, whenever the damping can be represented as weak coupling to a large reservoir.
Editorial inference: the symmetrization strategy suggests a general recipe—approximate a weakly non-Hermitian generator by a Hermitian one via a single scaling parameter $\eta$—that could be tested independently on small Carleman systems before relying on the full oscillator reduction.
Editorial inference: because the harmonic oscillator case is BQP-complete, the paper's claims, if correct, imply that the entire forced, time-dependent, and nonlinear family is classically hard to simulate even though each reduction step is only polynomial-time.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First, the forced-oscillator reduction is a genuine contribution. Embedding the forcing into a larger conservative system, with the auxiliary mass as a perturbation parameter, is clean and the complexity statement for that section is plausible. If the paper only claimed that, I'd take it seriously.

But the nonlinear and time-dependent results all hang on the symmetrization of the Carleman system, and that step is broken. Section 5.1 defines the diagonal blocks as A_i = -i H_1 (times projectors), which are anti-Hermitian. Lemma 32 then asserts Q has Hermitian diagonal blocks, which contradicts its own setup, and applies Lemma 33 to the generator H in Eq. (175). Lemma 33 requires H Hermitian so that e^{-iHt} is norm-preserving. With anti-Hermitian diagonals and unpaired off-diagonal B blocks, H is not Hermitian and e^{-iHt} is not unitary. The symmetrized operator Q(η) in Eq. (165) therefore also fails to be Hermitian, so the QSVT Hamiltonian simulation step in Theorem 44 has no valid target. Since Theorem 44 feeds every later theorem (15, 16, 18, 20, 22), the main claims do not follow from the proofs.

The paper is not careless: the Carleman truncation error analysis is standard and properly cited, the non-resonance conditions are stated honestly, and the authors acknowledge limitations in the conclusion. The problem is that the one novel technical tool—symmetrization—does not work as written. A possible repair would be to redefine the Carleman embedding so the diagonal blocks are Hermitian, or to use a block-encoding of the genuine non-Hermitian Q and accept the associated costs. But as it stands, the claim that nonlinear Schrödinger dynamics can be symmetrized into a Hermitian evolution is unsupported.

Minor issue: Theorem 44 asserts one can always rescale the equation so βη² < 1, but rescaling changes H1 and H2; that is not free and needs a proof.

Who should read it: researchers in quantum simulation of classical dynamics will want the forced-oscillator construction. The nonlinear results need major repair before they can be relied on.

My recommendation: send it to peer review, because the forced-oscillator contribution is worth refereeing and the symmetrization might be fixable. But the referee should be told to focus on Lemma 32, and the nonlinear theorems should not be accepted in the current form.

Referee Report

4 major / 5 minor

Summary. The paper proposes quantum algorithms for simulating forced, nonlinear, and time-dependent coupled classical oscillators. The strategy is to reduce each classical system to a nonlinear Schrödinger equation, linearize that equation by Carleman embedding, "symmetrize" the resulting non-Hermitian linear system into a Hermitian one, and simulate the Hermitian system with QSVT-based Hamiltonian simulation. The main complexity claims are Theorem 13 for forced oscillators, Theorems 15 and 16 for the nonlinear Schrödinger equation, and Theorems 18, 20, and 22 for nonlinear, time-dependent-stiffness, and time-dependent-forced oscillator systems. All of these except Theorem 13 rely on the symmetrization engine of Theorem 44, which in turn relies on Lemmas 32 and 33.

Significance. If correct, the results would substantially extend the exponential speedup of Babbush et al. for coupled harmonic oscillators to non-conservative, time-dependent, and weakly nonlinear classical systems, with complexity polynomial in the number of masses and almost linear in evolution time. The paper contains a considerable amount of explicit construction: oracle definitions, block-encoding lemmas, and reduction chains from each classical system to a nonlinear Schrödinger equation. I also note that the paper relies on external convergence results for Carleman linearization (Refs. [17,39]) and standard Hamiltonian simulation machinery, so I do not see a circularity problem. However, the central symmetrization argument is invalid as written, and that invalid argument supports essentially all of the paper's new claims; the forced-oscillator theorem is the only part that may stand independently, and even that requires a separate check of the perturbation analysis in Section 4.

major comments (4)

[§5.1, Eq. (143) and definitions of A_i, B_i] The Carleman system in Eq. (143) is written as d|p>/dt = -i Q |p> with first block row [H1, iH2, 0, ...]. However, Section 5.1 then defines A1 = (|0><0|)^{⊗k-1} ⊗ (-iH1) and B1 = (|0><0|)^{⊗k-2} ⊗ (|0>⊗H2). Substituting these into Eq. (152) gives a top-block equation d|ψ>/dt = -H1|ψ> - i H2|ψ⊗ψ>, which differs from the target equation (6), d|ψ>/dt = -iH1|ψ> + H2|ψ⊗ψ>. This sign and factor-of-i inconsistency propagates into every subsequent use of the Carleman embedding and makes the stated reduction from a nonlinear Schrödinger equation to a linear system incorrect.
[§5.3, Lemmas 32 and 33 and Eq. (175)] Lemma 32 applies Lemma 33 with H = ∑_l (|l><l|⊗A_{l+1} + |l><l+1|⊗B_{l+1}) in Eq. (175). Lemma 33 assumes H is block diagonal with Hermitian blocks, which is used to conclude that e^{-iHt} is norm-preserving and to obtain Eq. (182). The H in Eq. (175) is not block diagonal: it contains off-diagonal B_{l+1} blocks, and under the Section 5.1 definitions the A blocks are anti-Hermitian rather than Hermitian. Thus the premise of Lemma 33 fails, and the bound η ≥ sqrt(||hat H||) (1 + ||p(0)||/ϵ) t_s is unsupported. Since Theorem 44 invokes Lemma 32, and Theorems 15, 16, 18, 20, and 22 invoke Theorem 44, the main complexity claims of the paper do not follow from the presented proof.
[§5.3, Lemma 32 and definition of D|hat p>] With the stated definitions, |hat w_i> = η^{-(k-i)}|w_i> and D = diag(η^{k-1}I, ..., I), we have D|hat p> = ∑_i |i-1>⊗|w_i> = |p> exactly. Therefore the claimed approximation ∥D|hat p> - |p>∥ ≤ ϵ is an identity, not a convergence statement. The subsequent error bound in Theorem 44, written as ∥T|φ> - |p>∥ ≤ ϵ with T = ℵD, is then also vacuous because T|φ> = D|hat p> = |p> by construction. The symmetrization error needs to be defined with an internally consistent scaling of the Carleman variables; as written, the proof does not establish any approximation.
[§5.5, Theorem 44 and projection probability] Theorem 44 states that the probability of projecting onto the first Carleman block is p1 ∈ O((⟨ψ(t)|ψ(t)⟩/β)(1-βη²)) and adds that 'one can always rescale the nonlinear Schrödinger equation to ensure βη² < 1'. This rescaling claim is not justified in the proof: rescaling |ψ> also changes the norm of H2 and hence the Carleman truncation order and the definition of η through Eq. (220). Since βη² < 1 is an input condition for the stated probability bound, the proof needs to show that the rescaling is compatible with the other assumptions of the theorem, and it does not.

minor comments (5)

[Throughout] The name 'Carlemann' is consistently misspelled; the standard spelling is 'Carleman'.
[§5.3, Lemma 34] Lemma 34 writes hat H_1 = i|1><0|⊗B_1† and hat H_2 = i|2><1|⊗B_2†, but Lemma 32 defines hat H without the factor i. The factor i should be reconciled between the two statements.
[§4.3, Theorem 13] The proof of Theorem 13 contains an unclear phrase 'requireG =O(...)' with a missing space, and Proposition 29's derivation of E/E_sys contains the dimensionally inconsistent expression O(N f_max^2 l^3 t/ϵ^2); these need clarification.
[§4.1, Eq. (26)] Eq. (26) writes M¨x = K0x + γK'x, while the oscillator definitions in Section 2 use M¨x = -Kx. The sign conventions for K0 and K' should be stated consistently, because the perturbation result Theorem 25 is stated in terms of K0 and K'.
[§5.4, Lemmas 41 and 42] Lemma 41 constructs a block encoding of A_i = (|0><0|)^{⊗k-i}⊗(-i(⊕_{i-1}H1)), while Lemma 42 cites the same block-encoding construction U(⊕_{i-1}(-iH1)) for the B_i matrix, which should instead involve H2. This appears to be a copy-and-paste error and makes the stated block-encoding constants unreliable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain rests on external Carleman error bounds and standard Hamiltonian simulation results, not on fitted parameters or self-referential premises.

full rationale

The paper's claimed reductions are self-contained against independent benchmarks. The forced-oscillator result (Theorem 13) is built from an explicit embedding into a larger conservative oscillator system (Theorem 27) with perturbation bounds (Theorem 25) whose auxiliary initial conditions are chosen to match the forcing terms, not fitted after the fact. The nonlinear-Schrodinger results (Theorems 15, 16, 18, 20, 22) all rest on Theorem 44, whose Carleman truncation order is inherited from external convergence analyses [17, 39] and whose Hamiltonian simulation cost uses standard QSVT/block-encoding machinery [18]. The symmetrization parameter eta is chosen by a sufficient error bound (Lemma 32/33), not tuned to reproduce the final complexity claim. The only self-citation ([8], used to motivate impossibility of general nonlinear quantum dynamics) is not load-bearing for any theorem. A separate mathematical-soundness concern exists: Lemma 32 invokes Lemma 33 with an H that appears non-Hermitian under the printed A_i definitions, and Theorem 44 states without proof that one can always rescale so that beta*eta^2 < 1. Those are correctness risks, not circular reductions: they do not equate an output with an input by definition, and no fitted parameter is renamed as a prediction. Accordingly, no circular step can be exhibited and the appropriate score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central claims rest on two imported truncation-error results, a set of domain assumptions about the input class, and an unproven norm-preservation condition for the nonlinear Schrodinger mapping. The accounting shows that the contribution is a reduction chain, not a derivation of new classical dynamics.

free parameters (3)

Auxiliary mass m_f (gamma = 1/m_f) = Chosen large, e.g., m_f >= t ||K'|| ||T|| ||M^{-1/2}|| (1 + Xi(0)/epsilon)
Hand-picked to suppress perturbation error in the forced-oscillator embedding; affects energy normalization and gate counts but is not fitted to data.
Symmetrization scale eta = sqrt(||H2|| k(k+1)/2 (1 + beta(1-beta^k)/((1-beta)epsilon)) t)
Algorithmic parameter set by the error bound in Lemma 32; chosen by hand from the desired accuracy rather than fitted to an external dataset.
Rescaling of initial conditions to make beta < 1 = beta < 1 after rescaling by a constant
Theorem 20 assumes that initial conditions can be rescaled with 1/E so that the norm parameter beta is below one; this is a normalization choice used to force convergence conditions.

assumptions (6)

domain assumption Oracle access model for matrix elements, initial state preparation, and sparse access (Definitions 4, 6, 8, 11).
All complexity statements are relative to these oracles; without them the reductions cannot be implemented.
domain assumption Time-dependent forces and stiffnesses admit finite Fourier expansions f_i(t)=sum_j f_ij cos(omega_ij t + phi_ij) and k_ij(t)=sum_l alpha_ij,l cos(omega_ij,l t + phi_ij,l).
Stated in Definitions 3, 9, and 10; this avoids Fourier truncation error but restricts the input class.
domain assumption Masses are positive and stiffness matrices are positive semidefinite for all t.
Definitions 2 and 9; needed for the oscillator interpretation and for the norm bounds used in the perturbation analysis.
ad hoc to paper Norm-non-increasing condition Re(<psi|H2|psi tensor psi>) <= 0 for the nonlinear Schrodinger equation.
Lemma 30 requires this condition; the paper does not prove that the oscillator reductions in Sections 6 to 8 satisfy it.
domain assumption Non-resonance gap Delta > 0 and R_r < 1.
Conditions in Theorems 15, 18, 20, and 22; imported from reference [39] to make the Carleman truncation error small over long times.
standard math External Carleman truncation error bounds from [17] and Theorem 1.1 of [39].
Used to set the truncation order k in Theorems 15, 16, 18, 20, and 22; these bounds are not re-derived in this paper.

invented entities (1)

Auxiliary and fictitious masses added to the oscillator system
purpose: Emulate time-dependent external forces and time-dependent stiffness through conservative couplings in a larger oscillator system (Theorem 27 and Theorem 55).
These are computational ancillas, not claimed physical degrees of freedom; no experimental signature is provided.

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Cite this review

Pith. "Pith review of Simulating Time Dependent and Nonlinear Classical Oscillators through Nonlinear Schr\"odingerization." pith.science (2026). https://pith.science/paper/T4MXNBJV

@misc{pith2026250517170,
  author       = {Pith},
  title        = {Pith review of: Simulating Time Dependent and Nonlinear Classical Oscillators through Nonlinear Schr\"odingerization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T4MXNBJV}},
  note         = {Machine review of arXiv:2505.17170}
}

abstract

We present quantum algorithms for simulating the dynamics of a broad class of classical oscillator systems containing $2^n$ coupled oscillators (Eg: $2^n$ masses coupled by springs), including those with time-dependent forces, time-varying stiffness matrices, and weak nonlinear interactions. This generalization of the Harmonic oscillator simulation algorithm is achieved through an approach that we call ``Nonlinear Schr\"{o}dingerization'', which involves reduction of the dynamical system to a nonlinear Schr\"{o}dinger equation and then reduced to a time-independent Schrodinger Equation through perturbative techniques. The linearization of the equation is performed using an approach that allows the dynamics of a nonlinear Schr\"odinger equation to be approximated as a linear Schr\"odinger equation in a higher dimensional space. This allows Hamiltonian Simulation algorithms to be applied to simulate the dynamics of resulting system. When the properties of the classical dynamical systems can be efficiently queried, and when the initial state can be efficiently prepared, the complexity of our quantum algorithm is polynomial in $n$, and almost linear in evolution time for most dynamical systems. Our work extends the applicability of quantum algorithms to simulate the dynamics of non-conservative and nonlinear classical systems, addressing key limitations in previous approaches.

Figures

Figures reproduced from arXiv: 2505.17170 by the authors.

**Figure 2.** Equivalence between a higher dimensional oscillator system and a simple oscillator with [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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