Efficient synthesis of universal Repeat-Until-Success circuits

Recently, it was shown that Repeat-Until-Success (RUS) circuits can achieve a 2.5 times reduction in expected depth over ancilla-free techniques for single-qubit unitary decomposition. However, the previously best-known algorithm to synthesize RUS circuits requires exponential classical runtime. In this work we present an algorithm to synthesize an RUS circuit to approximate any given singlequbit unitary within precision ε in probabilistically polynomial classical runtime. Our synthesis approach uses the Clifford+T basis, plus one ancilla qubit and measurement. We provide numerical evidence that our RUS circuits have an expected T -count on average 2.5 times lower than the theoretical lower bound of 3 log2(1/ε) for ancilla-free single-qubit circuit decomposition.