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

TL;DR

This paper introduces a quantum circuit synthesis method based on Repeat-Until-Success that efficiently performs approximate arithmetic operations like multiplication and reciprocals with fewer qubits than traditional methods.

Contribution

The authors present a novel quantum synthesis approach for functions using RUS circuits, significantly reducing qubit requirements for arithmetic operations.

Findings

RUS circuits enable approximate multiplication and reciprocals with only 3 ancilla qubits.

The method requires fewer qubits than conventional quantum arithmetic approaches.

It offers a quantum arithmetic framework that differs from standard algorithmic techniques.

Abstract

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.