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(T2)), 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(egT/2) over a specified family of single-qubit initial states, where g>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.
Related
- Sparse Carleman Embeddings for Nonlinear Discrete Dynamical Systems
- OPINE: Orthogonal Projection in Invertible Nonlinear Embeddings for Turbulent Flows
- On the numerical limitations of dual Koopman von Neumann embeddings for solving conservative nonlinear ordinary differential equations on quantum computers
- On the sample complexity of the generalized hidden shift problem over arbitrary finite groups
- Quantization and Mirror Reduction Do Not Commute in Hamiltonian Embeddings of Nonreciprocal Dynamics
BibTeX
@misc{ito2026quantum,
title = {{Quantum Complexity of Ancilla-Free Unitary Embeddings for Nonlinear Dynamics via Generalized State-Dependent Double-Bracket Flows}},
author = {Yuki Ito and Hideaki Hakoshima and Keisuke Fujii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37802},
eprint = {2609.37802},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37802},
}
RIS
TY - UNPB
TI - Quantum Complexity of Ancilla-Free Unitary Embeddings for Nonlinear Dynamics via Generalized State-Dependent Double-Bracket Flows
AU - Yuki Ito
AU - Hideaki Hakoshima
AU - Keisuke Fujii
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37802
UR - https://arxiv.org/abs/2609.37802
AB - 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^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^gT/2) over a specified family of single-qubit initial states, where g>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.
ER -