Fig. 4: Extension to a three-dimensional gauge field.

a, b Evolutions of the state vector in the parameter space with two sequential operations \({U}_{{\theta }_{1},{\theta }_{2}}^{{\prime} }\) and \({U}_{{\phi }_{1},{\phi }_{2}}^{{\prime} }\). c, d Evolutions of the state vector in the parameter space with two sequential operations \({U}_{{\phi }_{1},{\phi }_{2}}^{{\prime} }\) and \({U}_{{\theta }_{1},{\theta }_{2}}^{{\prime} }\). Red (yellow) arrows denote the initial state Ψ_i (final state Ψ_f). e–g Theoretical calculation results of commutation of the two evolution operators \(\left[{U}_{{{{\rm{CW}}}}},{U}_{{{{\rm{CCW}}}}}\right]\) versus parameters of the applied gauge field θ₁, ϕ₁ and ϕ₂ for e θ₂ = 0, f θ₂ = π/2, and g θ₂ = π, respectively. The red dots, lines, and surface represent regions where \(\left[{U}_{{{{\rm{CW}}}}},{U}_{{{{\rm{CCW}}}}}\right]=0\) (i.e., an Abelian gauge field). The colormaps are the slices of \({S}^{{\prime} }\), where \({S}^{{\prime} }\ne \, 0\) represents the genuine non-Abelian cases.