The fundamental structure of quantum algorithms for finding the roots of a polynomial function

The Abel–Ruffini theorem (also known as Abel’s impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here we solve partly the mathematical problem by finding the roots of the following example polynomial function $$f(x)=(x+2.5)(x-3.5)(x-4.5)(x-6.5)(x-8.5)$$ without knowing the five roots. We first propose necessary and sufficient conditions for root-finding problem. These are the essences of quantum algorithms for finding the roots of a polynomial function $$f(x)=x^m +a_{m-1}x^{m-1}+...+a_1x+ a_0$$ . As a result, we find a simple relation between the root-finding problem and the oracle. Let r be a root. Then $$f(r) =0$$ ; thus, the oracle becomes the identity operator I. That is, $$U_f=U_0=I$$ if f is zero. In more detail, the phase kickback does not occur if f is zero. The phase kickback occurs if f is not zero. Here all the roots are in the real numbers $$\textbf{R}$$ . All the roots are different numbers, and the number of the roots is m. The essence of our deep desire to describe our honest view is a new insight into root-finding problem for future studies over the academic world as we discover straightly the importance of the phase kickback in the root-finding problems.