  In periodically driven (Floquet) systems, evolution typically results in an
infinite-temperature thermal state due to continuous energy absorption over
time. However, before reaching thermal equilibrium, such systems may
transiently pass through a meta-stable state known as a prethermal state. This
prethermal state can exhibit phenomena not commonly observed in equilibrium,
such as discrete time crystals (DTCs), making it an intriguing platform for
exploring out-of-equilibrium dynamics. Here, we investigate the relaxation
dynamics of initially prepared product states under periodic driving in a
kicked Ising model using the IBM Quantum Heron processor, comprising 133
superconducting qubits arranged on a heavy-hexagonal lattice, over up to $100$
time steps. We identify the presence of a prethermal regime characterised by
magnetisation measurements oscillating at twice the period of the Floquet cycle
and demonstrate its robustness against perturbations to the transverse field.
Our results provide evidence supporting the realisation of a period-doubling
DTC in a two-dimensional system. Moreover, we discover that the longitudinal
field induces additional amplitude modulations in the magnetisation with a
period incommensurate with the driving period, leading to the emergence of
discrete time quasicrystals (DTQCs). These observations are further validated
through comparison with tensor-network and state-vector simulations. Our
findings not only enhance our understanding of clean DTCs in two dimensions but
also highlight the utility of digital quantum computers for simulating the
dynamics of quantum many-body systems, addressing challenges faced by
state-of-the-art classical simulations.
