Quantum brachistochrone problem for two spins-1/2 with anisotropic Heisenberg interaction

We study the quantum brachistochrone evolution for a system of two spins-$\frac{1}{2}$ described by an anisotropic Heisenberg Hamiltonian without $zx$, $zy$ interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by $\mathrm{\parallel }\uparrow \uparrow ⟩$, $\mathrm{\parallel }\downarrow \downarrow ⟩$ and $\mathrm{\parallel }\uparrow \downarrow ⟩$, $\mathrm{\parallel }\downarrow \uparrow ⟩$ separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an entangler gate, SWAP gate, iSWAP gate et al.