Scope of fast-forwarding for quantum nonlinear differential-equation algorithms

Determine to what extent quantum algorithms for nonlinear differential equations, beyond the weakly nonlinear dissipative cases treated with fast-forwarded time complexity, can be fast-forwarded.

Background

The paper develops a fast-forwarded quantum algorithm for weakly nonlinear dissipative ordinary differential equations by combining Carleman embedding with a fast-forwarded linear-combination-of-Hamiltonian-simulation method. The resulting complexity is explicitly independent of the total evolution time under dissipativity and weak-nonlinearity assumptions.

The authors note that fast-forwarding is substantially less developed for nonlinear differential equations than for Hamiltonian simulation and linear differential equations. Their numerical experiments suggest possible extensions to systems with stronger nonlinearities and to certain non-resonant systems, but they do not establish a general characterization of which nonlinear differential equations admit fast-forwarded quantum algorithms.

References

So far it is not established if this quadratic cost reduction of the dissipative component is possible for mixed oscillatory-dissipative systems.

— Quantum Simulation of Dissipative Non-Markovian Coupled Classical Oscillators  (2609.20721 - Schade et al., 17 Sep 2026) in Section 7.3, “Fast-Forwarding”

However, there have been fewer works on fast-forwarding quantum nonlinear differential equation algorithms, and it remains unclear to what extent such algorithms can be fast-forwarded.

— Fast-forwarding quantum algorithms for weakly nonlinear dissipative differential equations and beyond  (2608.25822 - Li et al., 26 Aug 2026) in Section 1, Introduction

Extending this idea to non-resonant ODEs with time-dependent coefficients may require new techniques, as the Carleman convergence has not been shown in this case, and the contour-integral-based quantum algorithm in principle only applies to functions of time-independent matrices.

— Fast-forwarding quantum algorithms for weakly nonlinear dissipative differential equations and beyond  (2608.25822 - Li et al., 26 Aug 2026) in Section 6, Conclusion