Repeat-Until-Success (RUS) circuits are compiled in a Clifford+T basis by selecting a suitable cyclotomic integer approximation of a target rotation so that the rotation is approximated within a predetermined precision. The cyclotomic integer approximation is randomly modified until a modified value can be expanded into a single-qubit unitary matrix by solving one or more norm equations. The matrix is then expanded into a two-qubit unitary matrix of special form, which is then decomposed into an optimal two-qubit Clifford+T circuit. A two-qubit RUS circuit using a primary qubit and an ancillary qubit is then obtained based on the latter decomposition. An alternate embodiment is disclosed that keeps the total T-depth of the derived circuit small using at most 3 additional ancilla qubits. Arbitrary unitary matrices defined over the cyclotomic field of 8th roots of unity are implemented with RUS circuits.