We present a non-deterministic circuit decomposition technique for approximating an arbitrary single-qubit unitary to within distance epsilon that requires significantly fewer non-Clifford gates than deterministic decomposition techniques. We develop “Repeat-Until-Success” (RUS) circuits and characterize unitaries that can be exactly represented as an RUS circuit. Our RUS circuits operate by conditioning on a given measurement outcome and using only a small number of non-Clifford gates and ancilla qubits. We construct an algorithm based on RUS circuits that approximates an arbitrary single-qubit Z-axis rotation to within distance epsilon, where the number of T gates scales as 1.3 log(1/epsilon) – 2.79, an improvement of roughly three-fold over state-of-the-art techniques. We then extend our algorithm and show that a scaling of 2.4 log(1/epsilon) – 3.3 can be achieved for arbitrary unitaries and a small range of epsilon, which is roughly twice as good as optimal deterministic decomposition methods.