  When simulating the time evolution of quantum many-body systems on a digital
quantum computer, one faces the challenges of quantum noise and of the Trotter
error due to time discretization. The Trotter error in integrable spin chains
can be under control if the discrete time evolution preserves integrability. In
this work we implement, on a real quantum computer and on classical simulators,
the integrable Trotterization of the spin-1/2 Heisenberg XXX spin chain. We
study how quantum noise affects the time evolution of several conserved
charges, and observe the decay of the expectation values. We in addition study
the early time behaviors of the time evolution, which can potentially be used
to benchmark quantum devices and algorithms in the future. We also provide an
efficient method to generate the conserved charges at higher orders.
