  Improving the understanding of strongly correlated quantum many body systems
such as gases of interacting atoms or electrons is one of the most important
challenges in modern condensed matter physics, materials research and
chemistry. Enormous progress has been made in the past decades in developing
both classical and quantum approaches to calculate, simulate and experimentally
probe the properties of such systems. In this work we use a combination of
classical and quantum methods to experimentally explore the properties of an
interacting quantum gas by creating experimental realizations of continuous
matrix product states - a class of states which has proven extremely powerful
as a variational ansatz for numerical simulations. By systematically preparing
and probing these states using a circuit quantum electrodynamics (cQED) system
we experimentally determine a good approximation to the ground-state wave
function of the Lieb-Liniger Hamiltonian, which describes an interacting Bose
gas in one dimension. Since the simulated Hamiltonian is encoded in the
measurement observable rather than the controlled quantum system, this approach
has the potential to apply to exotic models involving multicomponent
interacting fields. Our findings also hint at the possibility of experimentally
exploring general properties of matrix product states and entanglement theory.
The scheme presented here is applicable to a broad range of systems exploiting
strong and tunable light-matter interactions.
