Efficient synthesis of universal Repeat-Until-Success circuits

TL;DR

This paper introduces a probabilistically polynomial-time algorithm for synthesizing Repeat-Until-Success circuits that approximate single-qubit unitaries efficiently, reducing T-count compared to previous methods.

Contribution

It presents the first efficient classical algorithm for synthesizing RUS circuits with provable approximation guarantees and reduced T-count using minimal ancilla qubits.

Findings

Expected T-count is on average 2.5 times lower than the theoretical lower bound.

The algorithm achieves approximation within precision ε in probabilistically polynomial runtime.

Numerical evidence supports the efficiency and effectiveness of the proposed synthesis method.

Abstract

Recently, it was shown that Repeat-Until-Success (RUS) circuits can achieve a $2.5$ times reduction in expected $T$-count 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 paper we present an algorithm to synthesize an RUS circuit to approximate any given single-qubit unitary within precision $\varepsilon$ 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 $3 \log_2 (1/\varepsilon)$ for ancilla-free single-qubit circuit decomposition.