  Many promising applications of quantum computing with a provable speedup
center around the HHL algorithm. Due to restrictions on the hardware and its
significant demand on qubits and gates in known implementations, its execution
is prohibitive on near-term quantum computers. Aiming to facilitate such
NISQ-implementations, we propose a novel circuit approximation technique that
enhances the arithmetic subroutines in the HHL, which resemble a particularly
resource-demanding component in small-scale settings. For this, we provide a
description of the algorithmic implementation of space-efficient rotations of
polynomial functions that do not demand explicit arithmetic calculations inside
the quantum circuit. We show how these types of circuits can be reduced in
depth by providing a simple and powerful approximation technique. Moreover, we
provide an algorithm that converts lookup-tables for arbitrary function
rotations into a structure that allows an application of the approximation
technique. This allows implementing approximate rotation circuits for many
polynomial and non-polynomial functions. Experimental results obtained for
realistic early-application dimensions show significant improvements compared
to the state-of-the-art, yielding small circuits while achieving good
approximations.
