The exact synthesis of 1- and 2-qubit Clifford+T circuits

Abstract: We describe a new method for the decomposition of an arbitrary $n$ qubit operator with entries in $\mathbb{Z}[i,\frac{1}{\sqrt{2}}]$, i.e., of the form $(a+b\sqrt{2}+i(c+d\sqrt{2}))/{\sqrt{2}^{k}}$, into Clifford+$T$ operators where $n\le 2$. This method achieves a bound of $O(k)$ gates using at most one ancilla using decomposition into $1$- and $2$-level matrices which was first proposed by Giles and Selinger.