Many-Objective Optimisation of Global Quantum Internet Topologies: A Compact Parametric Representation with an Analytical BDCZ Oracle.

We address the design of global quantum internet topologies that jointly optimise four physical objectives: average end-to-end Werner-state fidelity, entanglement rate, latency and coverage, evaluated with an analytical BDCZ oracle whose path selection is exact under the stated Werner-composition model and whose satellite links respect a single-satellite visibility limit. The search space is heterogeneous (fibre, satellite, free-space optical), comprising 1002=4950 possible edges over 100 cities; direct per-edge categorical optimisation degenerates. We propose a mixed representation of ten physical design parameters plus one stochastic realisation index (eleven decision variables in total) decoding into a topology via k-nearest-neighbour rules. Seven many-objective algorithms (NSGA-II, NSGA-III, MOEA/D, SMS-EMOA, RVEA, SPEA2, AGE-MOEA-II) are compared at a matched budget over 30 seeds, scored against a shared reference front. NSGA-II attains the highest mean hypervolume (0.353±0.040) but is statistically indistinguishable from SPEA2 and AGE-MOEA-II (0.352, 0.347); after Holm correction across all 63 tests, this top group separates significantly from NSGA-III, SMS-EMOA, RVEA and MOEA/D, though NSGA-III and SMS-EMOA remain indistinguishable in HV/IGD (spacing unresolved). NSGA-II's lead is driven by 6 of its 30 seeds reaching a higher-HV region, a bimodality we report. An eight-variant decoder ablation (NSGA-III) shows fibre is necessary while satellite's contribution is not detectable, and one classical constructor (a Random Geometric Graph) remains non-dominated; none of the 561 front points connects all 100 cities under the satellite model. On four additional real-city instances (five algorithms), MOEA/D is last throughout; SMS-EMOA/RVEA ranks vary by instance, and NSGA-II/NSGA-III reverses, without significance, twice; the fibre transmittance law is cross-checked against SeQUeNCe (Spearman ρ=1.0). The pooled front is released as QI-Bench-Pareto-v1.