  The Quantum Rabi model serves as a pivotal theoretical framework for
elucidating the nuanced interplay between light and matter. Utilizing circuit
quantum electrodynamics on a chip, we address the challenge of achieving deep
strong coupling in Quantum Cavity Electrodynamics (cQED). Despite progress in
superconducting circuits and trapped ions, experimental realization has been
limited to spectroscopy. Our focus is on a transformative digital quantum
simulation, employing Trotterization with an augmented number of steps to
deconstruct a complex unitary Hamiltonian. This approach showcases the benefits
of digital techniques within superconducting circuits, offering universality,
flexibility, scalability, and high fidelity. Our goal is to demonstrate deep
strong coupling in cQED and understand the advantages of digital methods,
particularly in coherent measurement during time evolution with varying photon
counts in resonators. This opens avenues to leverage quantum mechanics for
overcoming hardware limitations.
