Binary Optimization of Measurement Groupings for Quantum Energy Estimation

Abstract

Repeated measurements can dominate the resources required for quantum energy estimation, making the choice of which Pauli observables to measure together a central optimization problem for variational quantum algorithms. We formulate fully commuting measurement grouping as a classical binary optimization problem based on clique selection and construct non-overlapping groups using mixed-integer linear programming (MILP). For a benchmark of molecular Hamiltonians, groupings optimized with approximate covariances reduce the non-overlapping measurement requirement $\varepsilon^2M$ by $51.8\%$ on average relative to sorted insertion (SI). Using the MILP groups to initialize iterative coefficient splitting (ICS), denoted MILP-ICS, yields an average $24.3\%$ reduction relative to ICS initialized from SI (SI-ICS). The optimized groups also transfer across nearby molecular geometries while preserving substantial measurement savings. We further introduce O-clique, which directly optimizes overlapping commuting supports through candidate-clique selection and coefficient profiles. Although O-clique provides only modest additional reductions beyond MILP-ICS, it reduces the measurement requirement by $27.3\%$ on average relative to SI-ICS with 100 iterations, despite using only five final coefficient refinement iterations, indicating improved support quality. Finally, we extend the comparison to Fermi--Hubbard, Kitaev--Heisenberg--$\Gamma$, and XYZ lattice Hamiltonians, where MILP-based groupings substantially outperform the corresponding SI-based strategies. Together, these results show that variance-informed optimization of measurement-group structure can substantially reduce sampling costs across molecular and lattice Hamiltonians, with optimized non-overlapping groups providing strong, transferable initializations and direct overlapping optimization offering further gains.