  Loading functions into quantum computers represents an essential step in
several quantum algorithms, such as quantum partial differential equation
solvers. Therefore, the inefficiency of this process leads to a major
bottleneck for the application of these algorithms. Here, we present and
compare two efficient methods for the amplitude encoding of real polynomial
functions on $n$ qubits. This case holds special relevance, as any continuous
function on a closed interval can be uniformly approximated with arbitrary
precision by a polynomial function. The first approach relies on the matrix
product state representation. We study and benchmark the approximations of the
target state when the bond dimension is assumed to be small. The second
algorithm combines two subroutines. Initially we encode the linear function
into the quantum registers with a shallow sequence of multi-controlled gates
that loads the linear function's Hadamard-Walsh series, exploring how
truncating the Hadamard-Walsh series of the linear function affects the final
fidelity. Applying the inverse discrete Hadamard-Walsh transform transforms the
series coefficients into an amplitude encoding of the linear function. Then, we
use this construction as a building block to achieve a block encoding of the
amplitudes corresponding to the linear function on $k_0$ qubits and apply the
quantum singular value transformation that implements a polynomial
transformation to the block encoding of the amplitudes. This unitary together
with the Amplitude Amplification algorithm will enable us to prepare the
quantum state that encodes the polynomial function on $k_0$ qubits. Finally we
pad $n-k_0$ qubits to generate an approximated encoding of the polynomial on
$n$ qubits, analyzing the error depending on $k_0$. In this regard, our
methodology proposes a method to improve the state-of-the-art complexity by
introducing controllable errors.
