# 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 $$\frac{1}{2^N}\bigotimes_{i=1}^N \left[I+xX_i+\sqrt{1-x^2}Z_i\right]$$

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!

The authors might have meant that cross-terms like $$x \sqrt{1-x^2}$$ can improve the ability of the quantum circuit to approximate $$f$$ beyond what could be done with just an $$N$$-th order polynomial.

Using the authors' notation we can write the input state as $$\rho(x) = \sum_k a_k(x) P_k$$ with $$P_k \in \{I, X, Y, Z\}^N$$. Specifically, $$a_I(x) = 1$$, $$a_X(x) = x$$, and $$a_Z(x) = \sqrt{1-x^2}$$, and we define $$a_{PP'}(x) = a_P(x) a_{P'}(x)$$. After applying a parameterized unitary $$U(\theta)$$ and measuring an arbitrary observable we'll have an output of the form $$\hat{f}(x) = \sum_{k } b_{k}(\theta) a_k(x) \tag{1}$$

where $$b_k(\theta)$$ absorbs the components of the observable as well as the action of $$U(\theta)$$. For simplicity lets assume the coefficients $$b_k$$ can be arbitrary. An $$N$$-th order polynomial has the form

$$g_N(x) = c_0 + c_1 x + c_2 x^2 + \cdots + c_N x^N \tag{2}$$

If we take the case where $$N=2$$, $$g_2(x) = c_0 + c_1 x + c_2 x^2$$ but Eq. (1) contains terms like $$a_{XZ}(x) = x\sqrt{1-x^2}$$ and $$a_{IZ} = \sqrt{1-x^2}$$. Writing $$b_k := b_k(\theta)$$ and rewriting subscripts $$I$$, $$X$$, $$XZ$$, as integers for brevity, the circuit therefore produces functions of the form

\begin{align} \hat{f}(x) &= b_0 + b_1 x + b_2 x^2 + b_3 \sqrt{1-x} + b_4 x\sqrt{1-x^2}\tag{3} \\&= b_0 + b_1 x + b_2x^2 + b_3 \left(1 - \frac{1}{2} x^2 - \frac{1}{8} x^4 +\cdots \right) + b_4 \left(x - \frac{1}{2} x^3 - \frac{1}{8} x^5 +\cdots \right) \tag{4} \\&= (b_0 + b_3) + (b_1 + b_4)x + \left(b_2 - \frac{b_3}{2}\right) x^2 -\frac{b_4}{2}x^3 - \frac{b_3}{8} x^4 + \cdots \tag{5} \end{align}

where I've used the Maclaurin series expansion for $$\sqrt{1-x^2}$$. So this function is strictly more expressive than Eq. (2) as we can recover an arbitrary second order polynomial $$g_2(x)$$ by choosing $$\theta$$ such that $$b_3=b_4=0$$ and $$b_k = c_k$$ otherwise. Conversely, keeping these additional terms might greatly improve the convergence rate (with respect to $$N$$) of $$\hat{f}$$ for approximating certain functions compared to $$g_N$$ (e.g. when $$f(x)$$ contains terms like $$\sqrt{1-x^2}$$).