Title:Mutually unbiased bases as extremal probes of isotropic random Hamiltonians

Abstract:Finite structured sets are frequently used to probe high-dimensional energy landscapes, yet their extreme-value performance depends on correlations between sampled energies rather than on marginal coverage alone. We study this question for the Gaussian field generated by an isotropic random traceless Hamiltonian. In every dimension admitting a complete system of mutually unbiased bases (MUBs), we prove that the union of the complete MUB system maximizes the expected field maximum among all labeled unions of the same number, \(d+1\), of orthonormal bases. More strongly, its maximum is largest in stochastic order. The mechanism is explicit: every orthonormal basis induces the same regular-simplex Gaussian block, while mutual unbiasedness makes all cross-block covariances vanish and hence makes the blocks independent. A centered-convex Gaussian correlation inequality then shows that this independent-block arrangement is extremal. The isotropic ensemble is used as a structure-free reference model and is not asserted to represent generic local, combinatorial, or chemistry Hamiltonians. We derive a radial corollary and an exact collapse result showing that a fully matched MUB construction has no family dependence for diagonal costs. Two finite-size variational case studies are consequently interpreted only as tests outside the exact isotropic model: a composite MUB-XRot warm-start ansatz and a local family search for a non-diagonal quantum relaxation. Both change solution quality in selected instances but require larger classical-search budgets and do not establish asymptotic trainability or a runtime advantage.