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!


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