Quantum arithmetic and numerical analysis using Repeat-Until-Success circuits

Understand

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.