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

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