  A single-qubit circuit can approximate any bounded complex function stored in
the degrees of freedom defining its quantum gates. The single-qubit approximant
is operated through a series of gates that take as their input 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 result of this quantum circuit becomes
more accurate as the number of re-uploadings of the independent variable
increases. In this work, we provide two different proofs stating that a
single-qubit circuit is a universal approximant, first by a direct casting of a
series of exponentials to standard Fourier analysis and, second, by an
analogous proof for quantum systems of the universal approximation theorem for
neural networks. We also benchmark the performance of both methods and compare
them to 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.
