Efficient Synthesis of Universal Repeat-Until-Success Quantum Circuits

Recently it was shown that the resources required to implement unitary operations on a quantum computer can be reduced by using probabilistic quantum circuits called repeat-until-success (RUS) circuits. However, the previously best-known algorithm to synthesize a RUS circuit for a given target unitary requires exponential classical runtime. We present a probabilistically polynomial-time algorithm to synthesize a RUS circuit to approximate any given single-qubit unitary to precision \(\epsilon\) over the \(\text{Clifford}+T\) basis. Surprisingly, the \(T\) count of the synthesized RUS circuit surpasses the theoretical lower bound of \(3 {log}_2(1/\epsilon )\) that holds for purely unitary single-qubit circuit decomposition. By taking advantage of measurement and an ancilla qubit, RUS circuits achieve an expected \(T\) count of \(1.15 {log}_2(1/\epsilon )\) for single-qubit \(z\) rotations. Our method leverages the fact that the set of unitaries implementable by RUS protocols has a higher density in the space of all unitaries compared to the density of purely unitary implementations.