Repeat-Until-Success: Non-deterministic decomposition of single-qubit unitaries

We present a decomposition technique that uses non-deterministic circuits to approximate an arbitrary single-qubit unitary to within distance ε and requires significantly fewer non-Clifford gates than existing 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 ε, where the number of T gates scales as 1.26log₂(1/ε) - 3.53, an improvement of roughly three-fold over state-of-the-art techniques. We then extend our algorithm and show that a scaling of 2.4log₂(1/ε) - 3.28 can be achieved for arbitrary unitaries and a small range of ε, which is roughly twice as good as optimal deterministic decomposition methods.