The integrable Lie-Trotter-Suzuki decomposition for nonlinear dynamics: Efficient Hamiltonian simulation of the Stepanoff flow

We propose an efficient time integration strategy for nonlinear dynamics: the integrable Lie-Trotter-Suzuki (LTS) decomposition [1].  First, express the 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. This ensures that the full unitary evolution operator can be represented as a product of terms that each have an efficient explicit tensor product factorization that can be handled efficiently with tensor network and/or quantum block-encoding methods.  We then apply this method 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 modeled using a numerical emulator of fault-tolerant quantum computers, and the results are found to be consistent with classical simulations. In general, challenges lie 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 decompositions due to the topology of the flow [2].