  Quantum simulation, fundamental in quantum algorithm design, extends far
beyond its foundational roots, powering diverse quantum computing applications.
However, optimizing the compilation of quantum Hamiltonian simulation poses
significant challenges. Existing approaches fall short in reconciling
deterministic and randomized compilation, lack appropriate intermediate
representations, and struggle to guarantee correctness. Addressing these
challenges, we present MarQSim, a novel compilation framework. MarQSim
leverages a Markov chain-based approach, encapsulated in the Hamiltonian Term
Transition Graph, adeptly reconciling deterministic and randomized compilation
benefits. We rigorously prove its algorithmic efficiency and correctness
criteria. Furthermore, we formulate a Min-Cost Flow model that can tune
transition matrices to enforce correctness while accommodating various
optimization objectives. Experimental results demonstrate MarQSim's superiority
in generating more efficient quantum circuits for simulating various quantum
Hamiltonians while maintaining precision.
