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Quantum simulation can be implemented in pure digital or analog ways, each with their pros and cons. By taking advantage of the universality of a digital route and the efficiency of analog simulation, hybrid digital-analog approaches can enrich the possibilities for quantum simulation. We use a unique hybrid approach to experimentally perform a quantum simulation of phase-controlled dynamics resulting from a closed-contour interaction (CCI) within certain multi-level systems in superconducting quantum circuits. Due to symmetry constraints, such systems cannot host an inherent CCI. Nevertheless, by assembling analog modules corresponding to their natural evolutions and specially designed digital modules constructed from standard quantum logic gates, we can bypass such constraints and realize an effective CCI in these systems. Based on this realization, we demonstrate a variety of related and interesting phenomena, including phase-controlled chiral dynamics, separation of chiral enantiomers, and a new mechanism to generate entangled states based on CCI.