Improvement of quantum walks search algorithm in single-marked vertex graph

Abstract Quantum walks are powerful tools for building quantum search algorithms. However, these success probabilities far below 1. Amplitude amplification is usually used to amplify probability, but the souffl'e problem follows. Only stop at right step can we achieve a maximum probability. Otherwise, as number of steps increases, probability may decrease, which will cause troubles in practical application algorithm when optimal unknown. 
 In this work, define generalized interpolated instead amplitude amplification, not only improve also avoid problem. We choose special case and construct series new algorithms based on phase estimation fast-forwarding, respectively. Especially, by combining our with both reduce time complexity from $\Theta((\varepsilon^{-1})\sqrt{\Heg})$ $\Theta(\log(\varepsilon^{-1})\sqrt{\Heg})$ ancilla qubits required $\Theta(\log(\varepsilon^{-1})+\log\sqrt{\Heg})$ $\Theta(\log\log(\varepsilon^{-1})+\log\sqrt{\Heg})$, where $\varepsilon$ denotes precision $\Heg$ classical hitting time. addition, show that construction stationary states corresponding reversible Markov chains.
 Finally, give an slowly evolving chain sequence applying walks, necessary premise adiabatic state preparation.