In this series of talks, we plan to learn basics and esstentials of mathematics in quantum circuits, following the Chapter 4 of the “bible”: Chuang-Nielson, Quantum Computation and Quantum Information, 10 th Anniversary Edition. We will continue to discuss different meaning and importance of universal quantum gates by the means of approximation of unitary operators in quantum circuits. Furthermore, we will give detailed proof of the Solovay–Kitaev theorem.