  Quantum algorithms frequently rely on truncated series approximations,
necessitating a high truncation order to achieve even moderate accuracy and
consequently resulting in intensive circuit complexity. In response, we propose
a general framework, the Randomized Truncated Series (RTS), which offers two
avenues for simplifying circuits: a quadratic improvement on the truncation
error and enabling a continuously adjustable effective truncation order. The
core idea is that the random mixing of two series of specific forms generates a
substantial reduction in the truncation error. We present an error analysis for
RTS with a new mixing lemma accounting for near-unitary operators. To
demonstrate the effectiveness and versatility of RTS, we provide four
illustrative examples within the context of Linear Combination of Unitary,
Quantum Signal Processing, and solving Quantum Differential Equations. RTS shed
light on the path towards practical quantum advantage.
