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Fast-forwarding quantum algorithms for weakly nonlinear dissipative differential equations and beyond

Yixiang Li, Dong An

arXiv:2608.25822v1quant-phmath.NA

TL;DR

Large nonlinear differential equations pose scaling challenges, while quantum nonlinear-ODE algorithms commonly retain evolution-time dependence. The paper combines Carleman embedding with LCHS for weakly nonlinear dissipative ODEs, obtaining complexity explicitly independent of T and exploring broader numerical regimes.

  • Problem

    Existing quantum nonlinear differential-equation algorithms often scale linearly or worse in evolution time T, limiting long-time simulation, while fast-forwarding applicability remains unclear.

  • Method

    The algorithm combines Carleman embedding for time-dependent coefficients with LCHS applied to the resulting linearized ODEs, while simplifying post-selection.

  • Results

    The proposed algorithm solves weakly nonlinear dissipative ODEs with complexity independent of T, and numerical studies identify possible fast-forwarding beyond the weakly nonlinear regime.

  • Takeaways & Limitations

    Fast-forwarded scaling may extend in practice to dissipative ODEs with stronger nonlinearity and to non-resonant systems under additional stability conditions.

  • Takeaways & Limitations

    For tested non-resonant systems, the linearized ODEs are not dissipative, so the paper’s fast-forwarded algorithm does not apply without additional conditions or a different contour-integral algorithm.

Abstract

from arXiv · show

We study a fast-forwarded quantum algorithm for solving weakly nonlinear dissipative ordinary differential equations. Our approach is a combination of the Carleman embedding technique and the linear combination of Hamiltonian simulation algorithm for linearized systems with fast-forwarded scaling. The complexity of our algorithm does not explicitly depend on the evolution time $T$, thus greatly improving the previous state-of-the-art $\widetilde{\mathcal{O}}(\sqrt{T})$ to $\mathcal{O}(1)$, and any remaining time dependence enters through the output norm and forcing parameters. We rigorously analyze the performance of this approach by convergence guarantees of the Carleman embedding for time-dependent coefficient matrices and detailed complexity estimates, and improve the realization of the Carleman-embedding-based algorithms by simplifying the post-selection step. In addition, we perform a numerical study on differential equations beyond the weakly nonlinear case, and identify possibility of achieving fast-forwarding scaling for systems with stronger nonlinearity or linear non-resonant effect.

1 Introduction

The paper targets fast-forwarded quantum simulation of weakly nonlinear dissipative ODEs, addressing the evolution-time scaling that limits long-time prediction. It combines Carleman embedding with LCHS, simplifies post-selection, and numerically examines broader regimes.

  • Large nonlinear differential-equation systems challenge classical integrators because complexity can grow polynomially or exponentially with dimension or spatial resolution.
  • Most existing quantum nonlinear differential-equation algorithms scale linearly or worse in evolution time T, limiting long-time simulations and asymptotic predictions.
  • Fast-forwarding nonlinear differential equations remains less developed than fast-forwarding Hamiltonian and linear differential-equation simulations, with its applicability still uncertain.
  • Under dissipative linear dynamics and relatively weaker nonlinear terms, the proposed algorithm solves the ODE with complexity explicitly independent of T.
  • Carleman embedding followed by LCHS yields fast-forwarded scaling through convergence analysis for time-dependent coefficients and strict dissipativity of the linearized system.
  • Discarding the ancilla register replaces prior projection onto the degree-one sector, while controlled higher-order embedding errors preserve a valid numerical solution.
  • Numerics suggest stronger-than-predicted nonlinearities may still permit convergence and practical fast-forwarding for dissipative systems, while non-resonant systems require additional conditions or another algorithm.

2 Setup

The setup focuses on quadratic nonlinear ODEs whose linear component is dissipative and whose nonlinear component is relatively weak. The framework also extends to Lyapunov-stable systems through a positive-definite change of variables.

  • The considered quadratic nonlinear ODEs require the linear coefficient matrix F1(t) to have uniformly negative logarithmic norm.
  • The weak-nonlinearity setting additionally constrains the nonlinear contribution relative to linear dissipation.
  • Lyapunov-stable ODEs can replace the logarithmic-norm condition with a time-independent positive-definite matrix P satisfying F1(t)†P + PF1(t) ≤ −2cP.
  • After the transformation ũ(t) = Qu(t), the theoretical and fast-forwarded results extend when F̃1(t) has uniformly negative logarithmic norm, the associated R parameter is bounded by 1, and a block-encoding of Q is available.

3 Carleman embedding with time-dependent coefficients

The paper linearizes time-dependent quadratic ODEs by rescaling the solution and applying Carleman embedding, then truncates the lifted hierarchy to obtain a finite linear system. A rescaling choice supports dissipativity and convergence bounds for the truncation error.

  • Carleman embedding for time-dependent coefficients linearizes the nonlinear ODE and provides explicit parameter-dependent convergence analysis.
  • Rescaling introduces uγ(t) = γu(t) and transformed coefficients so that additional technical conditions needed for Carleman convergence can be imposed.
  • The lifted state contains tensor powers of the rescaled solution, including uγ(t), uγ(t)⊗2, and higher-order terms, yielding an infinite-dimensional linear ODE.
  • Truncating the lifted hierarchy at order N produces a finite linearized ODE with state ȳ = [ȳ1; ȳ2; · · · ; ȳN] and a specified tensor-power initial condition.
  • The Carleman error vector η is bounded by combining Duhamel’s principle with the coefficient-matrix and rescaled-solution norm bounds.
  • The rescaling factor γ is selected as the smaller root of a convergence condition, after which lemmas bound the coefficient matrix and the rescaled solution norm.

4 Fast-forwarded quantum algorithm

The algorithm linearizes weakly nonlinear dissipative ODEs with Carleman embedding, then solves the enlarged system using fast-forwarded linear-combination-of-Hamiltonian-simulation techniques. Under the theorem’s assumptions, its complexity is explicitly independent of evolution time T, with residual dependence through output and forcing norms.

  • Input access models: The method assumes coherent time-dependent block-encodings of coefficient matrices and a state-preparation oracle for the initial condition.The query complexity is measured primarily by calls to these oracles.
  • Algorithm: Carleman embedding converts the quadratic nonlinear ODE into an enlarged linear dissipative system.The lifted state contains u(t) and higher-order tensor powers of the rescaled solution.
  • Algorithm: Fast-forwarded linear ODE simulation uses truncated Duhamel evolution, linear-combination Hamiltonian simulation, and linear-combination-of-unitaries discretization.Dissipation permits a truncation time T0 independent of sufficiently large T, subject to dissipation strength and target accuracy.
  • Post-processing: Discarding the appropriate ancilla or higher-order registers extracts the normalized nonlinear solution without the usual first-component post-selection.The resulting reduced state approximates the target density matrix through a partial trace.
  • Guarantees: Theorem 4.1 guarantees an approximation of |u(T)⟩⟨u(T)| with trace distance at most ϵ under dissipativity and weak nonlinearity assumptions.The construction balances Carleman and numerical errors at O(ϵ).
  • Complexity: The overall complexity has no explicit T dependence, while implicit dependence remains through ∥u(T)∥ and ∥F0∥L1(T−T0,T).The output-norm dependence is attributed to post-selection requirements, and forcing dependence may be bounded independently of T in suitable cases.

5 Numerical results

The numerical study tests Carleman convergence and lifted-system dissipativity across dissipative, conservative, and non-resonant weakly nonlinear ODE scenarios. It also examines whether the fast-forwarded solver can apply to the resulting linearized systems.

  • Experimental design: The experiments vary the Carleman truncation order N and compare the first lifted block against numerically computed reference solutions.Reference solutions use the DOP853 integrator in scipy.
  • Experimental design: The study evaluates fast-forwarding eligibility by computing the dissipativity margin of each truncated Carleman matrix.A positive margin δN indicates a dissipative lifted system whose solution cost can be independent of T.

5.1 Dissipative systems

For tested dissipative systems, Carleman truncation converges for both theoretically weak and numerically stronger nonlinearities, while every tested lifted system remains dissipative.

  • Setting: The tested parameter set includes both R < 1 and R > 1 cases, with R approximately {0.56, 1.01, 1.16, 0.40, 0.68, 1.57}.The experiments use the original equations without rescaling.
  • Numerical observations: The truncation error decreases monotonically with N for all tested parameter choices, dropping by several orders of magnitude from N = 1 to N = 10 when R < 1.This rapid convergence agrees with the theoretical prediction.
  • Numerical observations: R > 1 cases, including R approximately 1.01, 1.16, and 1.57, also converge, but more slowly than R < 1 cases.These results suggest R < 1 is sufficient rather than a sharp necessary threshold for this model.
  • Numerical observations: The dissipativity margin δN remains positive and uniformly bounded away from zero across tested truncation orders and parameter choices.Thus the lifted systems remain dissipative and eligible for the fast-forwarded quantum ODE algorithm.

5.2 Conservative systems

In the tested partially conservative system, Carleman truncation accurately approximates the nonlinear trajectory, but the lifted linear system is not dissipative and therefore falls outside the fast-forwarded solver’s guaranteed setting.

  • Setting: The homogeneous partially conservative model is studied to T = 3, with an unchanged first entry of u and an analytically checkable reference solution.The DOP853 reference differs from the exact solution by 1.026×10^-11 in the reported supremum norm.
  • Numerical observations: The truncation error decreases rapidly and reaches the reference-solver accuracy floor by N = 9 or N = 10.This indicates strong numerical convergence of the Carleman approximation.
  • Numerical observations: The dissipativity margin satisfies δ1 = 0 and δN < 0 for all N ≥ 2, becoming more negative as N increases.The associated lifted linear system is therefore not dissipative.
  • Numerical observations: Carleman convergence and dissipativity of the lifted system are distinct properties in this conservative example.Accurate trajectory approximation does not imply decay of the lifted linear flow or applicability of the dissipative fast-forwarded solver.

5.3 Non-resonant systems

For non-resonant systems, Carleman embedding can converge beyond the sufficient condition R∆<1, but the truncated lifted system is not dissipative in the tested cases, preventing the preceding fast-forwarded algorithm.

  • 5.3.2 Numerical observations: R∆<1 is sufficient but not necessary: cases near 1.04, 2.09, 3.20, 4.40, and 5.71 also converge numerically.For sufficiently large R∆, truncation quality deteriorates substantially and eventually fails in the tested range.
  • 5.3.2 Numerical observations: δ1 = 0 and δN < 0 for N ≥2 across tested f2 values, with negativity strengthening as N and f2 increase.The truncated lifted linear system therefore fails to be dissipative in all tested cases.
  • 5.3.2 Numerical observations: Convergence of the Carleman embedding can occur without decay of the truncated lifted flow in the non-resonant setting.Thus, non-resonance helps explain approximation behavior but does not guarantee dissipativity.
  • 5.3.3 Implication on fast-forwarding quantum algorithms: Because the tested non-resonant linearized ODEs are non-dissipative even when Carleman embedding converges, the fast-forwarded algorithm described earlier does not apply.A contour-integral-based alternative is sketched under time-independent coefficients and eigenvalues of F1 with negative real parts.
  • 5.3.3 Implication on fast-forwarding quantum algorithms: The contour-integral alternative approximates e^(AT) using resolvents, quantum linear-system solvers, and LCU, with complexity independent of T under the stated homogeneous assumptions.The bound uses |e^(zT)| ≤ 1 on the chosen contour.

6 Conclusion

The paper proposes a fast-forwarded quantum algorithm for weakly nonlinear dissipative ODEs and extends the study numerically beyond that regime. It identifies broader settings where practical fast-forwarding may be possible, while leaving rigorous extensions open.

  • 6 Conclusion: The proposed algorithm solves weakly nonlinear dissipative ODEs with complexity independent of the evolution time.Its construction combines Carleman embedding for time-dependent coefficients with fast-forwarded LCHS for the linearized system.
  • 6 Conclusion: The work proves Carleman-embedding convergence for time-dependent coefficient matrices, analyzes performance, and simplifies the post-selection step.The conclusion also reports numerical study beyond the dissipative case.
  • 6 Conclusion: Numerical experiments identify the possibility of practical fast-forwarding for nonlinear ODEs beyond the weakly nonlinear dissipative setting.The broader possibilities include stronger nonlinearity or non-resonant effects, within the numerical scope studied.
  • 6 Conclusion: A complete rigorous analysis for contour-integral fast-forwarding of broader non-resonant systems remains future work.The proposed extension is sketched for homogeneous, time-independent systems and may require new techniques for time-dependent coefficients.

A Bounds of the Carleman error

The appendix develops Carleman-error bounds by adapting the time-independent proof strategy to time-dependent coefficients. It combines norm estimates, comparison arguments, and matrix bounds to control the embedding error.

  • A Bounds of the Carleman error: The appendix adapts the established time-independent Carleman-error proof strategy to time-dependent coefficients with explicit parameter dependence.The resulting analysis is organized through several supporting lemmas.
  • A Bounds of the Carleman error: The dissipativity estimate reduces to bounding the largest eigenvalue of the Hermitian part H(G) of an auxiliary matrix G.Gershgorin’s circle theorem is then applied to estimate that eigenvalue.
  • A Bounds of the Carleman error: The proof parameterizes the bounds with a = ∥F2∥, b = −µ(F1), c = ∥F0∥, r = a + c − b, and x = ∥u(0)∥.These quantities enter the rescaling and subsequent error estimates.
  • A Bounds of the Carleman error: A comparison ODE shows that the solution norm remains bounded by its initial norm under the condition R < 1.The argument introduces v−, v+, and w(t) to compare the original dynamics with a scalar equation.
  • A Bounds of the Carleman error: The Carleman error vector η is bounded using Duhamel’s principle together with the preceding solution-norm estimates.The estimate uses the rescaling relation involving γ, µ(F1), F2, F0, and R.
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