In this paper Quantum Circuit Learning it wrote that the ability of a quantum circuit to approximate a function can be enhanced by terms like $x\sqrt{1-x^2}$ ($x\in[-1,1])$. Given inputs {x,f(x)}, assume we are going to approximate an analytical function by a polynomial with higher terms up to the $n$th order, the steps are similar to the following:
1.construct a polynomial like $(1+x+\sqrt{1-x^2})^n$
2.apply a parameterized unitary transformation $U(\theta)$, and turn the polynomial into something like $(u_{i1}+u_{i2}x+u_{i3}\sqrt{1-x^2})^n$
3.minimize the cost function by tuning the parameters $\theta$ iteratively.
I am a little confused about how can terms like $x\sqrt{1-x^2}$ enhance the ability to approximate the function. Maybe it's implemented by introducing nonlinear terms, but I can't find the exact mathematical representation.
Thanks for any help in advance!