  We develop a method for approximate synthesis of single--qubit rotations of
the form $e^{-i f(\phi_1,\ldots,\phi_k)X}$ that is based on the
Repeat-Until-Success (RUS) framework for quantum circuit synthesis. We
demonstrate how smooth computable functions $f$ can be synthesized from two
basic primitives. This synthesis approach constitutes a manifestly quantum form
of arithmetic that differs greatly from the approaches commonly used in quantum
algorithms. The key advantage of our approach is that it requires far fewer
qubits than existing approaches: as a case in point, we show that using as few
as $3$ ancilla qubits, one can obtain RUS circuits for approximate
multiplication and reciprocals. We also analyze the costs of performing
multiplication and inversion on a quantum computer using conventional
approaches and find that they can require too many qubits to execute on a small
quantum computer, unlike our approach.
