The homogeneous Bethe-Salpeter equation (hBSE), describing a bound system in a genuinely relativistic quantum-field theory framework, was solved for the first time by using a D-Wave quantum annealer. After applying standard techniques of discretization, the hBSE, in ladder approximation, can be formally transformed in a generalized eigenvalue problem (GEVP), with two square matrices: one symmetric and the other non symmetric. The latter matrix poses the challenge of obtaining a suitable formal approach for investigating the non symmetric GEVP by means of a quantum annealer, i.e to recast it as a quadratic unconstrained binary optimization problem. A broad numerical analysis of the proposed algorithms, applied to matrices of dimension up to 64, was carried out by using both the proprietary simulated-anneaing package and the D-Wave Advantage 4.1 system. The numerical results very nicely compare with those obtained with standard classical algorithms, and also show interesting scalability features.