A quantum algorithm for a selected class of nonlinear differential equations has a theoretical query-complexity bound with no explicit dependence on how long the system evolves. The result applies to weakly nonlinear dissipative ordinary differential equations: systems whose linear part damps motion and whose nonlinear contribution is weaker than that damping under the paper’s assumptions.
That finding is narrower than a general claim about fast-forwarding nonlinear dynamics. The bound still depends indirectly on the size of the final solution and on the accumulated size of the forcing term during the final interval. The guarantee also assumes access to quantum oracles that coherently encode time-dependent coefficient matrices and prepare the initial state.
How the proposed solver works
The method first applies Carleman embedding and then works with an enlarged linear ordinary differential equation. The analysis derives a formal upper bound for the error from finite Carleman truncation after choosing the rescaling factor gamma as the smaller root of its defining equation.
The enlarged linear equation is handled with fast-forwarded linear-combination-of-Hamiltonian-simulation methods. After that calculation, the algorithm discards the added ancilla qubits to extract the state associated with the original variables. Under the stated conditions, and for sufficiently large evolution time and small target error, the output density matrix is claimed to lie within the requested trace-distance error of the normalized final solution.
The quantum implementation is measured primarily by the number of queries to the assumed oracles, so the stated complexity is tied to how the input data are supplied.
Numerical tests point beyond the formal guarantee
The numerical study compared finite Carleman truncations with reference trajectories produced by the DOP853 integrator in scipy. It examined dissipative, conservative and non-resonant ordinary differential-equation scenarios.
In every tested dissipative parameter setting, the truncation error fell monotonically as the truncation order N increased. Some cases in which the nonlinear-strength measure R was above the sufficient threshold of 1 still converged, although more slowly. The reported examples included values of R of about 1.01, 1.16 and 1.57, and the truncated lifted systems retained positive dissipativity margins that stayed uniformly away from zero.
These results suggest that the paper’s formal conditions are sufficient rather than a sharp boundary for the selected dissipative models. They do not establish that stronger nonlinearities will converge in general, because the beyond-threshold behavior was observed only numerically over the tested models and parameter choices.
Convergence did not guarantee the property needed for fast-forwarding
The conservative example made the distinction clearer. It was a homogeneous calculation run to T = 3. The DOP853 reference differed from the analytic solution by 1.026×10−11 in the reported supremum 2-norm, and the Carleman approximation reached that reference-solver accuracy floor by truncation order N = 9 or N = 10.
But the lifted system was not dissipative beyond the first truncation order: its margin was zero at N = 1 and negative for every N ≥ 2. In the paper’s framework, the example therefore shows that a Carleman approximation can converge accurately without supplying the dissipativity required by the fast-forwarded solver. The dissipative algorithm does not apply to that conservative case.
A second set of tests found the same tension
The non-resonant tests were also homogeneous and used T = 3, with the parameter f2 varied across the listed settings. The reported nonlinear-strength measure R∆ ranged from about 0.94 to 29.61. Several cases above the sufficient threshold still converged numerically, including examples near 1.04, 2.09, 3.2, 4.4 and 5.71, while sufficiently large values produced substantial deterioration and eventual failure in the tested range.
None of the tested non-resonant lifted systems met the dissipativity condition: the margin was zero at N = 1 and negative for N ≥ 2, with stronger negativity as N and f2 increased. Thus, some of these models showed Carleman convergence even though the dissipative fast-forwarding route was unavailable.
What remains unresolved
The analysis sketches a possible alternative based on contour integrals for a restricted non-resonant setting. The proposed route concerns homogeneous equations with time-independent coefficient matrices whose linear part has eigenvalues with negative real parts. It is presented as a heuristic pathway, not a complete rigorous performance guarantee, and broader inhomogeneous and time-dependent extensions remain open.
Taken together, the results support a specific theoretical conclusion: within the stated weakly nonlinear dissipative regime and oracle model, the algorithm’s accuracy guarantee removes explicit evolution-time dependence from the query bound. The simulations add evidence of convergence in selected cases outside that regime, but those observations remain model-specific and numerical.
The document is an arXiv version 1 preprint dated 26 August 2026.
Paper data and sources
Original title: Fast-forwarding quantum algorithms for weakly nonlinear dissipative differential equations and beyond
Authors: Yixiang Li, Dong An
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text