The approach to the employment of the mechanism of valence tautomerism (VT) for the design of molecular 2-qubit quantum gates has been extended to adducts 5–8 of Co^II bis-(malonates), bis-(acetylacetonates), bis-(hexafluoroacetylacetonates) and bis-(trifluoroacetylacetonates) with tetradentate tetraone (cyclic di-o-quinones) redox-active piperazine-2,3,5,6-tetraone L₅ (X = NH), 3,3,6,6-tetramethylcyclohexane-1,2,4,5-tetraone L₅ (X = C(CH₃)₂), cyclopenta[fg]acenaphthylene-1,2,5,6-tetraone L₆, cyclopenta[fg]acenaphthylene-3,4,7,8-tetraone L₇ and pyrene-4,5,9,10-tetraone L₈ and computationally studied using the B3LYP*/6-311++G(d,p) method. The calculations reveal ferromagnetic ordering of unpaired electrons in the low-spin electromeric forms of complexes 8 (R₁, R₂ = H, CH₃, CF₃) on the basis of L₈, which provides for the paramagnetic character of all three interconverting electromers of 8 and makes it possible to realize the two-step mechanism of thermally driven migration of paramagnetic centers between the pyrene-tetraone fragment and metal ions. Through the structural variation of the ancillary diketonate ligands the energy gaps between the electromeric forms of adducts 8 and energy barriers for their interconversion were adjusted to the range of values typical of thermal VT rearrangements. The best energy parameters for the occurrence of thermal two-step VT rearrangements involving all three paramagnetic electromeric forms can be achieved for complex 8 (R₁ = CH₃, R₂ = CF₃), which was shown to meet the principal conditions for compounds with a potential role of spin qubit carriers: thermal stability with respect to dissociation into the components, thermally accessible energy barriers for the interconversion of the electromers and weak coupling between their paramagnetic centers.