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I am a new beginner in quantum algorithms. Recently, I was trying to learn the quantum minimum finding algorithm appeared in here. I am struggling to fully understand the operations in the algorithm.

  1. What does the "Mark every item $j$ for which $T[j]<T[y]$" really mean? Does it mean we need to define a conditional phase flip oracle whose condition is to check whether $T[j]<T[y]$?
  2. Can such a conditional phase flip oracle be explicitly constructed?
  3. "Apply the quantum exponential searching algorithm of 2" is unclear to me. What type of quantum exponential searching algorithm should we use in [2]? Since we do not know how many items in a list satisfying the condition $T[j]<T[y]$, should we use the algorithm for unknown number of solutions? Which steps should we use in the searching algorithm?
  4. For step 2(b), "Apply the quantum exponential searching algorithm". Does it mean that one should apply the whole process of searching, e.g., the Grover algorithm, which applies the Grover operator $G$ several times, or use the amplitude amplification once?

Basically, I want to have a clear picture of the whole algorithmic process and make sure to understand how each step works.

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