Efficient explicit circuit for quantum state preparation of piecewise continuous functions

Efficiently uploading data into quantum states is essential for many quantum algorithms to achieve advantage across various applications. In this paper, we address this challenge by developing a method to upload a polynomial function $f(x)$ on the interval $x \in [-1,1]$ into a pure quantum state consisting of qubits, where a discretized $f(x)$ is the amplitude of this state. The preparation cost has $\mathcal{O}(n\log n)$ scaling in the number of qubits $n$ and linear scaling with the degree of the polynomial $Q$. This efficiency allows the preparation of states whose amplitudes correspond to high-degree polynomials (up to $10^4$), enabling accurate approximation of functions that admit efficient polynomial series representations and whose amplitude profiles are not extremely localized. We provide a fully explicit circuit realization, based on four real polynomials that meet specific parity and boundedness conditions. We extend this construction to cover piece-wise polynomial functions, a case not previously addressed explicitly in the literature, the algorithm scaling linearly with the number of piecewise parts. Our method achieves efficient quantum circuit implementation and we present detailed gate counting and resource analysis.