  Quantum Layout Synthesis (QLS) plays a crucial role in optimizing quantum
circuit execution on physical quantum devices. As we enter the era where
quantum computers have hundreds of qubits, we are faced with scalability issues
using optimal approaches and degrading heuristic methods' performance due to
the lack of global optimization. To this end, we introduce a hybrid design that
obtains the much improved solution for the heuristic method utilizing the
multilevel framework, which is an effective methodology to solve large-scale
problems in VLSI design. In this paper, we present ML-QLS, the first multilevel
quantum layout tool with a scalable refinement operation integrated with novel
cost functions and clustering strategies. Our clustering provides valuable
insights into generating a proper problem approximation for quantum circuits
and devices. Our experimental results demonstrate that ML-QLS can scale up to
problems involving hundreds of qubits and achieve a remarkable 52% performance
improvement over leading heuristic QLS tools for large circuits, which
underscores the effectiveness of multilevel frameworks in quantum applications.
