NMinimize[{Sqrt[1 - 4 Sqrt[3] b c d + d^2 - 5 c^2 (-1 + d^2) + 2 a (-Sqrt[2] b d - Sqrt[6] c (-1 + d^2))], a^2 + c^2 == 1 && b^2 + d^2 == 1}, {a, b, c, d}]