Physics Guided Generative Optimization for Trotter Suzuki Decomposition

Trotter Suzuki product formulas are the standard route to Hamiltonian evolution on noisy intermediate-scale quantum (NISQ) hardware, but their accuracy depends on three coupled choices: term grouping, product-formula order, and time-step allocation. Grouping and order are discrete, which makes direct gradient optimization infeasible and forces existing compilers to rely on static heuristics. We describe P-GONE, a method that combines a conditional diffusion model (D3PM + DDPM), a graph neural network (GNN) encoder, and closed-loop REINFORCE fine-tuning to jointly learn grouping, order, and time-step optimization over a mixed discrete-continuous space. Under fidelity-matched conditions (F ≥ 0.95), the method achieves circuit depth 86 versus 1673 for Qiskit fourth-order (ungrouped, Suzuki-4), about 19.4× compression, and 141 for Paulihedral (first-order Trotter), about 1.6× compression. At T=0.90 the method also beats the Qiskit group-commuting teacher (65 vs 103, 1.6× compression), though at T=0.95 the teacher still leads -- a stratified pattern that points toward fidelity-aware fine-tuning. Under a standard depolarizing noise model, the method achieves noisy fidelity roughly 2× the Qiskit fourth-order baseline (0.743 vs 0.380). Ablation shows a clear hierarchy: order learning > time allocation > grouping. Best-of-N sampling (N=32 is a practical sweet spot) and CFG guidance give flexible fidelity-depth trade-offs at inference. The method works well on structured Hamiltonians (TFIM, Heisenberg), but random Pauli Hamiltonians fail entirely at T ≥ 0.95 -- a boundary that defines where the method applies.