Quantum Complexity of Ancilla-Free Unitary Embeddings for Nonlinear Dynamics via Generalized State-Dependent Double-Bracket Flows

Abstract: Simulating nonlinear dynamics with quantum computers has gained increasing attention. In general, such simulations require additional quantum resources because unitary quantum evolution is linear. A fundamental question is how nonlinear dynamics can be embedded into fully coherent, ancilla-free unitary circuits and how the complexity of the dynamics governs the required quantum resources. In this work, we generalize the ancilla-free double-bracket quantum algorithm for imaginary-time evolution by replacing its state-independent Hamiltonian with a state-dependent Hermitian operator. Our framework recursively calls an initial state preparation oracle and its inverse, and prepares the target solution to any prescribed accuracy using a fully coherent, ancilla-free unitary embedding. We relate the query cost to the complexity of the nonlinear dynamics, specifically their sensitivity to initial conditions. We obtain query upper bounds of $\exp(O(T))$, $\exp(O(T<sup>2))$, and $\exp(\exp(O(T)))$ when the distance between solutions contracts at least exponentially (contractive), does not increase (nonexpansive), or grows at most exponentially (expansive), respectively, where $T$ is the target evolution time. For the discrete Gross--Pitaevskii equation, our ancilla-free double-bracket circuit achieves optimal worst-case query complexity $Θ(e<sup>{gT/2})$ over a specified family of single-qubit initial states, where $g&gt;0$ is the nonlinearity strength. These results connect the complexity of nonlinear dynamics to the query cost of coherent quantum simulation and provide a foundation for designing ancilla-free unitary embeddings with optimal query complexity.