  A single-qubit circuit can approximate any bounded complex function stored in
the degrees of freedom defining its quantum gates. The single-qubit approximant
presented in this work is operated through a series of gates that take as their
parameterization the independent variable of the target function and an
additional set of adjustable parameters. The independent variable is
re-uploaded in every gate while the parameters are optimized for each target
function. The output state of this quantum circuit becomes more accurate as the
number of re-uploadings of the independent variable increases, i. e., as more
layers of gates parameterized with the independent variable are applied. In
this work, we provide two different proofs of this claim related to both the
Fourier series and the Universal Approximation Theorem for Neural Networks, and
we benchmark both methods against their classical counterparts. We further
implement a single-qubit approximant in a real superconducting qubit device,
demonstrating how the ability to describe a set of functions improves with the
depth of the quantum circuit. This work shows the robustness of the
re-uploading technique on Quantum Machine Learning.
