We develop a method for approximate synthesis of single--qubit rotations ofthe form $e^{-i f(phi_1,ldots,phi_k)X}$ that is based on theRepeat-Until-Success (RUS) framework for quantum circuit synthesis. Wedemonstrate how smooth computable functions $f$ can be synthesized from twobasic primitives. This synthesis approach constitutes a manifestly quantum formof arithmetic that differs greatly from the approaches commonly used in quantumalgorithms. The key advantage of our approach is that it requires far fewerqubits than existing approaches: as a case in point, we show that using as fewas $3$ ancilla qubits, one can obtain RUS circuits for approximatemultiplication and reciprocals. We also analyze the costs of performingmultiplication and inversion on a quantum computer using conventionalapproaches and find that they can require too many qubits to execute on a smallquantum computer, unlike our approach.
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机译：我们基于量子电路的重复-直到成功（RUS）框架，开发了一种近似合成$ e ^ {-if（ phi_1， ldots， phi_k）X} $形式的单量子位旋转的方法合成。演示如何从两个基本原语合成平滑的可计算函数$ f $。这种合成方法构成了一种明显的量子算术形式，与量子算法中常用的方法大不相同。我们的方法的主要优点是它所需的量子位远少于现有方法：作为一个例子，我们证明了使用少至$ 3 $的辅助量子位，就能获得用于近似乘法和倒数的RUS电路。我们还分析了使用常规方法在量子计算机上执行乘法和求逆的成本，发现与我们的方法不同，它们可能需要太多的量子位才能在小型量子计算机上执行。
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