  Quantum simulations provide means to probe challenging problems within
controllable quantum systems. However, implementing or simulating deep-strong
nonlinear couplings between bosonic oscillators on physical platforms remains a
challenge. We present a deterministic simulation technique that efficiently and
accurately models nonlinear bosonic dynamics. This technique alternates between
tunable Rabi and SNAP gates, both of which are available on experimental
platforms such as trapped ions and superconducting circuits. Our proposed
simulation method facilitates high-fidelity modeling of phenomena that emerge
from higher-order bosonic interactions, with an exponential reduction in
resource usage compared to other techniques. We demonstrate the potential of
our technique by accurately reproducing key phenomena and other distinctive
characteristics of ideal nonlinear optomechanical systems. Our technique serves
as a valuable tool for simulating complex quantum interactions, simultaneously
paving the way for new capabilities in quantum computing through the use of
hybrid qubit-oscillator systems.
