A Physics-Grounded QUBO Encoding of Irrigation Scheduling for QAOA

Abstract

Rotational irrigation scheduling in water-scarce Central Asia is a densely coupled combinatorial problem: soil-moisture memory links each irrigation decision to all later days within a zone, field adjacency couples zones on overlapping window days, and rigid canal rotations quantize water delivery in time. We formulate it as a Quadratic Unconstrained Binary Optimization (QUBO) by linearizing the root-zone water balance, so the quadratic crop-stress objective generates the physical couplings as 2-local Ising interactions with no higher-order terms; only the water-budget constraint requires an artificial all-to-all penalty, which we certify with an instance-adaptive weight bound an order of magnitude tighter than generic prescriptions. Every instance is built from observed data for a cotton district in Khorezm, Uzbekistan: NASA POWER meteorology, FAO-56 Penman--Monteith evapotranspiration, SoilGrids~2.0 hydraulics, measured capillary fluxes, and documented canal-rotation windows that enter as qubit-count reductions. We benchmark four tiers -- exact solvers, matched-budget heuristics, ideal-statevector quantum approximate optimization algorithm (QAOA), and noise-model plus IBM Heron execution -- and add a scaling study on soil instances up to 584 variables. Exact branch-and-bound proves optimality in seconds through 150 variables, and heuristic degradation at fixed evaluation budget is repaired by scaling the budget, so no classical scalability wall appears at deployment-relevant sizes, and none is claimed. On hardware, the informative signal is optimum-sampling enrichment over uniform sampling. We claim no quantum advantage; we deliver a physically grounded, data-complete, reproducible encoding of a societally critical scheduling problem for the quantum-utility era.