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We construct bi-frequency solitary waves of the nonlinear Dirac equation withthe scalar self-interaction (the Soler model) and the Dirac--Klein--Gordon withYukawa self-interaction. These solitary waves provide a natural implementationof qubit and qudit states in the theory of quantum computing. We show therelation of $\pm 2\omega\mathrm{i}$ eigenvalues of the linearization at asolitary wave, Bogoliubov $\mathbf{SU}(1,1)$ symmetry, and the existence ofbi-frequency solitary waves. We show that the spectral stability of these wavesreduces to spectral stability of usual (one-frequency) solitary waves.

Spectral stability of bi-frequency solitary waves in Soler and Dirac--Klein--Gordon models