Mutually Unbiased Bases for Variational Quantum Initialization: Basis-Union Optimality and Adaptive Family Search

We study mutually unbiased bases (MUBs) as structured finite initialization and adaptation families for variational quantum algorithms. The main theoretical result is that, in every dimension admitting a complete set of MUBs, the complete MUB ensemble maximizes isotropic Gaussian random-Hamiltonian width among all unions of d+1 orthonormal bases in C^d. Equivalently, within this basis-union class, it gives the smallest expected best-of-set minimum for random-Hamiltonian minimization. The proof represents each orthonormal basis as a regular-simplex Gaussian block and uses a centered-convex Gaussian correlation inequality to show that the independent-block case, realized by complete MUBs, is stochastically extremal. We also record a radial extension for Hamiltonians H=RG with R nonnegative and independent, and the unrestricted qubit case, where complete qubit MUBs are globally optimal among arbitrary six-state ensembles by a Bloch-sphere/octahedron mean-width argument. We then separate this coverage theorem from variational training dynamics. For diagonal QUBO costs, the MUB-family dependence of a fully matched construction collapses; for the canonical b=0 label it reduces to ordinary X-mixer QAOA. The empirical method is therefore adaptive MUB-XRot warm-start QAOA rather than canonical matched-mixer MUB-QAOA. In a cross-problem benchmark over MaxCut, weighted MaxCut, MIS, weighted MIS, and knapsack, adaptive MUB-XRot is non-worse than standard QAOA in 80.0% of 1500 paired cases, with win/tie/loss 829/371/300 and mean decoded-ratio improvement +0.1616. A separate QRAO MaxCut study shows that bit-flip MUB-family search reaches mean relaxed ratio 0.921 and improves over the X-variational baseline by +0.0608. The evidence is quality-oriented and incurs substantial runtime overhead; no quantum-advantage claim is made.