Building generic gates from a restricted gate set is a difficult but important problem, especially in the noisy regime where only a limited set of noise-resistant gates are available, e.g., fault-tolerant Clifford gates (generated by Hadamard, phase, and cnot gates) and fault-tolerant Toffoli gates. The Toffoli is also often used as a building block of many algorithms and will need to be constructed if not directly available. This makes $\text{Clifford}+\text{Toffoli}$ an attractive gate set for building generic gates. In this Letter we give a simple and efficient algorithm for building an approximate single-qubit rotation using only $\text{Clifford}+\text{Toffoli}$, which in turn enables any generic single-qubit unitary. An important difference compared to earlier attempts is that the use of the Toffoli allows us to use simple rounding as opposed to a complicated approximation algorithm. The resulting gate array does not rely on repeatedly applying a fixed rotation, but immediately applies a rotation $R_{{\theta }^{*}}$ that is $\epsilon$-close to the desired rotation $R_{\theta }$, with a success probability strictly greater than $1/2$. It can be rerun upon failure, giving an expected number of repetitions strictly less than 2, an expected Toffoli count strictly less than $4\lceil log\frac{1}{\epsilon }\rceil$, an expected gate depth strictly less than $4\lceil log\frac{1}{\epsilon }\rceil +6$, and uses $2\lceil log\frac{1}{\epsilon }\rceil$ ancillas. The small circuit depth of our construction enables low-noise gates on existing quantum computational devices, and allows for arbitrary precision using a very modest number of ancillas.

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• Quantum circuits
• Quantum computation
• Quantum information processing