# Why does $x\sqrt{1-x^2}$ enhance the ability to approximate analytical functions in quantum circuit learning?

In this paper Quantum Circuit Learning they say 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 $$\{x,f(x)\}$$, it aims 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. Encoding $$x$$ by constructing a state $$\frac{1}{2^N}\otimes_{i=1}^N[I+xX_i+\sqrt{1-x^2}Z_i]^n$$

2. Apply a parameterized unitary transformation $$U(\theta)$$.

3. Minimize the cost function by tuning the parameters $$\theta$$ iteratively.

I am a little confused about how can terms like $$x\sqrt{1-x^2}$$ in the polynomial represented by the quantum state can enhance its 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!