Physics Guided Generative Optimization for Trotter Suzuki Decomposition

Product formulas for Trotter Suzuki simulation remain a practical route to Hamiltonian evolution on noisy intermediate scale quantum (NISQ) hardware, yet their accuracy hinges on three coupled choices: term grouping, product formula order, and timestep allocation. Toolchains such as Qiskit and Paulihedral lean on hand tuned heuristics, while the discrete nature of grouping and order makes naive gradient based optimization awkward. We describe a generate and evaluate loop: a conditional diffusion model proposes strategies, a physics informed neural network (PINN) supplies differentiable fidelity feedback, and a graph neural network (GNN) encodes commutator structure. Training spans a hybrid space (discrete grouping and order, continuous time steps); the closed loop uses REINFORCE and a Pareto tracker. On the transverse field Ising model (TFIM), under our primary comparison setup, the method reaches 85.6% of the fidelity of a fourth order Qiskit baseline (0.856) at roughly 21.8% of the circuit depth and 19.2% of the baseline CNOT count. Under an equal depth budget, fine tuning in the loop reached a best observed fidelity of 0.9994. Updated ablations show that, for a fixed training budget and default guidance knobs, module contributions depend on the training recipe and guidance hyperparameters CFG in particular needs to be tuned jointly with compute budget. Overall, the results suggest that "generative model and physics supervision" is a viable angle for NISQ oriented compilation, though where it wins still depends on the operating point.