We propose an efficient Lie-Trotter-Suzuki (LTS) decomposition time integration strategy for Koopman-von Neumann dynamics. First, express the Koopman-von Neumann generator, i.e. the advection operator, as a sum of integrable terms and then approximate the full evolution operator using an LTS decomposition of the desired order [1]. This integrable LTS decomposition ensures that the full unitary evolution operator can be represented as a product of terms that each have an efficient explicit tensor product factorization [2] that can be handled efficiently with tensor network and/or quantum block-encoding methods. We apply the I-LTS decomposition to an important non-integrable point example: the Stepanoff flow on the torus. We develop an efficient unitary quantum algorithm for simulating the Stepanoff flow as a classical version of a quantum map and provide a detailed description of the encoding of the two-dimensional system into a quantum circuit. The simulation is performed using a sequence of alternating quantum Fourier transform (QFT) and Quantum Signal Processing (QSP) circuits. The circuit is simulated using a numerical emulator of fault-tolerant quantum computers, and the results are found to be consistent with classical calculations [1]. In general, there is intrinsic complexity in the linear scaling of the advection operator with grid size and in the need to represent the evolution operator in different regions of phase space with different coordinate systems and/or integrable decompositions due to the topology of the flow.

The need to use multiple components or domains to achieve an integrable decomposition is intrinsically related to the intrinsic complexity encountered in the recently discovered globalized Carleman methods [3]. While the standard Carleman method fails to converge in regions where there are multiple fixed points, the globalized methods avoid this issue by shifting the basepoint and rescaling the domain of the linearized solution. This need for additional linearization charts arises from the interplay between different aspects of the problem: the need to globally maintain convergence, and, hence, analyticity of the solution, as well as the need to allow for trajectories to be attracted to nontrivial sets such as strange attractors. Tradeoffs in complexity between the global Carleman method and the Koopman-von Neumann method will be discussed.

References

[1] I. Joseph, I. Novikau, M. Montgomerey, J. Slawinska, D. Giannakis, “The integrable Lie-Trotter-Suzuki decomposition for nonlinear dynamics: Efficient Hamiltonian simulation of the Stepanoff flow,” to be submitted 2026.

[2] D. Giannakis, M. J. Latifi Jebelli, M. Montgomerey, P. Pfeffer, J. Schumacher, J. Slawinska, “Tensor network approximation of Koopman operators,” arXiv:2407.07242

[3] I. Novikau, I. Joseph, “Globalizing the Carleman linear embedding method for nonlinear dynamics,” arXiv:2510.15715