Optimal Stratification of a Sampling Frame: A Comparative Study of Classical, Quantum, and Quantum-Inspired Approaches
Optimal stratification aggregates strata into a small number of final strata to minimise total sample size required to meet target precision constraints. This combinatorial objective, reformulated as a within-cluster dispersion surrogate, can be expressed as a quadratic unconstrained binary optimisation (QUBO) problem. This paper reports a comparative case study of four solvers for that surrogate, run under heterogeneous free-tier constraints on an identical twenty-stratum frame from the swissmunicipalities dataset: a D-Wave-formulated QUBO, solved here by simulated annealing (QPU access unavailable); a gate-based quantum processor (IBM Quantum, running QAOA); a photonic entropy-quantum-computing device (QCi Dirac-3); and a classical GPU-based Ising machine as control (Fixstars Amplify AE). Solvers differ in hardware, computational budget, iteration count, and, for IBM, the encoding itself, so the comparison is a case study rather than a controlled experiment. Against a genetic-algorithm benchmark of 129 sample units, the best-known QUBO solution (objective 501.0) maps, after Bethel-Chromy evaluation, to a sample size of 160, a gap of roughly 24%, tied to the best solution found rather than a certified optimum. IBM and Dirac-3 fall further short, at 262 and 286; the IBM result is consistent with SWAP-routing degradation on a sparse qubit lattice, though not isolated from other causes, while Dirac-3, despite genuine optimisation, settles short of what a free GPU reaches in seconds. The paper concludes that a classical genetic algorithm remains, at the scale tested, the method of choice for optimal stratification, reflecting (a) limits of the dispersion surrogate versus the true Bethel-Chromy objective, not encodable as a low-degree polynomial, and (b) solver- and platform-specific limits in the IBM and Dirac-3 runs. These findings do not establish global optimality for either result.