Existence of an efficient quantum algorithm for piecewise Carleman embedding

Establish whether an efficient quantum algorithm can be realized for the non-adaptive piecewise Carleman embedding (PCE) method that simulates nonlinear ordinary differential equations by switching between Carleman charts when a trajectory crosses chart boundaries, and determine if such an algorithm can achieve quantum advantage despite the need to detect chart boundary crossings (potentially via measurements).

Background

The proposed globalized Carleman method uses piecewise switching of linearization charts to maintain convergence for nonlinear and chaotic systems. Implementing chart-boundary detection and chart transitions introduces nonlinear operations and measurements that challenge quantum implementations.

The authors suggest a hybrid classical–quantum approach but explicitly note that it is unresolved whether an efficient, fully quantum algorithm based on their piecewise technique can be achieved with quantum advantage.

References

However, the question of whether an efficient quantum algorithm based on the proposed piecewise technique can be realized remains open.

— Globalizing the Carleman linear embedding method for nonlinear dynamics  (2510.15715 - Novikau et al., 17 Oct 2025) in Section “Conclusion”

A further question is when optimal query complexity for simulating nonlinear dynamics can be attained by fully coherent, ancilla-free constructions.

— Quantum Complexity of Ancilla-Free Unitary Embeddings for Nonlinear Dynamics via Generalized State-Dependent Double-Bracket Flows  (2609.37802 - Ito et al., 29 Sep 2026) in Section 6, Conclusion and Discussion