  In classical computational chemistry, the coupled-cluster ansatz is one of
the most commonly used $ab~initio$ methods, which is critically limited by its
non-unitary nature. The unitary modification as an ideal solution to the
problem is, however, extremely inefficient in classical conventional
computation. Here, we provide the first experimental evidence that indeed the
unitary version of the coupled cluster ansatz can be reliably performed in
physical quantum system, a trapped ion system. We perform a simulation on the
electronic structure of a molecular ion (HeH$^+$), where the ground-state
energy surface curve is probed, energies of excited-states are studied and the
bond-dissociation is simulated non-perturbatively. Our simulation takes
advantages from quantum computation to overcome the intrinsic limitations in
classical computation and our experimental results indicate that the method is
promising for preparing molecular ground-states for quantum simulation.
