  For quantum state preparation, a non-unitary operator is typically designed
to decay undesirable states contained in an initial state using ancilla qubits
and a probabilistic action. Probabilistic algorithms do not accelerate the
computational process compared to classical ones. In this study, quantum
amplitude amplification (QAA) and multi-step probabilistic algorithms are
combined to achieve quadratic acceleration. This method outperforms quantum
phase estimation in terms of infidelity. The quadratic acceleration was
confirmed by the probabilistic imaginary-time evolution (PITE) method.
