Discrete Flow-Based Generative Models for Measurement Optimization in Quantum Computing

Achieving chemical accuracy in quantum simulations is often constrained by the measurement bottleneck: estimating operators requires a large number of shots, which remains costly even on fault-tolerant devices. Addressing this challenge involves a multi-objective optimization problem that balances the total shot count, the number of distinct measurement circuits, the total two-qubit gate count, and hardware-specific compilation constraints. Existing overlapping grouping methods, focused on reducing measurement counts, rely on an initial non-overlapping grouping of the Hamiltonian, generated through graph-coloring strategies or greedy heuristics to group commuting (FC) or qubit-wise-commuting Hamiltonian terms. We introduce an algorithm that adapts Generative Flow Networks (GFlowNets) to color graph representations of Hamiltonians, enabling the generation of reward-driven, non-overlapping groupings. Our approach samples colored graphs in proportion to a user-defined reward, allowing different objective terms to be incorporated into the reward, capturing multi-objective trade-offs. On benchmark molecular Hamiltonians, our method reduces measurement costs relative to sorted-insertion (SI) baselines and can reduce the two-qubit gate count for FC groupings. We show that the groupings generated with GFlowNets serve as better initializations for overlapping methods, particularly iterative coefficient splitting, further reducing measurement costs by 19\% on average for Jordan-Wigner-mapped Hamiltonians in FC groupings. Initializing overlapping methods with our groupings, which have lower two-qubit requirements, yields comparable reductions in measurement counts while preserving the two-qubit-count benefit. GFlowNets'generative policy framework not only reduces measurement and two-qubit gate costs but also provides flexibility for hardware-aware adaptations via its reward function.