  Quantum walks are powerful tools for building quantum search algorithms or
quantum sampling algorithms named the construction of quantum stationary state.
  However, the success probability of those algorithms are all far away from 1.
Amplitude amplification is usually used to amplify success probability, but the
souffl\'e problems follow. Only stop at the right step can we achieve a maximum
success probability. Otherwise, as the number of steps increases, the success
probability may decrease, which will cause troubles in practical application of
the algorithm when the optimal number of steps is not known.
  In this work, we define generalized interpolated quantum walks, which can
both improve the success probability of search algorithms and avoid the
souffl\'e problems.
  Then we combine generalized interpolation quantum walks with quantum
fast-forwarding. The combination both reduce the times of calling walk operator
of searching algorithm from $\Theta((\varepsilon^{-1})\sqrt{\Heg})$ to
$\Theta(\log(\varepsilon^{-1})\sqrt{\Heg})$ and reduces the number of ancilla
qubits required from $\Theta(\log(\varepsilon^{-1})+\log\sqrt{\Heg})$ to
$\Theta(\log\log(\varepsilon^{-1})+\log\sqrt{\Heg})$, and the souffle problem
is avoided while the success probability is improved, where $\varepsilon$
denotes the precision and $\Heg$ denotes the classical hitting time.
  Besides, we show that our generalized interpolated quantum walks can be used
to improve the construction of quantum states corresponding to stationary
distributions as well.
  Finally, we give an application that can be used to construct a slowly
evolving Markov chain sequence by applying generalized interpolated quantum
walks, which is the necessary premise in adiabatic stationary state
preparation.
