  We propose new formulations of max-sum and max-min dispersion problems that
enable solutions via the Grover adaptive search (GAS) quantum algorithm,
offering quadratic speedup. Dispersion problems are combinatorial optimization
problems classified as NP-hard, which appear often in coding theory and
wireless communications applications involving optimal codebook design. In
turn, GAS is a quantum exhaustive search algorithm that can be used to
implement full-fledged maximum-likelihood optimal solutions. In conventional
naive formulations however, it is typical to rely on a binary vector spaces,
resulting in search space sizes prohibitive even for GAS. To circumvent this
challenge, we instead formulate the search of optimal dispersion problem over
Dicke states, an equal superposition of binary vectors with equal Hamming
weights, which significantly reduces the search space leading to a
simplification of the quantum circuit via the elimination of penalty terms.
Additionally, we propose a method to replace distance coefficients with their
ranks, contributing to the reduction of the number of qubits. Our analysis
demonstrates that as a result of the proposed techniques a reduction in query
complexity compared to the conventional GAS using Hadamard transform is
achieved, enhancing the feasibility of the quantum-based solution of the
dispersion problem.
