t b a b a

lack box performs the following transformation
The oracle qubit can be initially in the state
| x〉 | q〉 ↦→ | x〉 | q ⊕ f(x)〉.
| q〉 = (1/ √ 2)(| 0〉− | 1〉).
If an input | x〉 is not a solution to our search problem, then the oracle qubit is unchanged.
On the other hand, a solution to the search problem transforms | 0〉 into | 1〉. Thus, the
transformation performed by the black box is
⎧
| x〉 (| 0〉− | 1〉)/ √ 2 ↦→
⎪⎨
⎪⎩
This transformation can be rewritten as
| x〉 (| 0〉− | 1〉)/ √ 2 if f(x) = 0 =⇒ Ex is not a solution
− | x〉 (| 0〉− | 1〉)/ √ 2 if f(x) = 1 =⇒ Ex is a solution.
O | x〉 (| 0〉− | 1〉)/ √ 2 = (−1) f(x) | x〉 (| 0〉− | 1〉)/ √ 2.
The state of the oracle qubit does not change and can be omitted from the description of
the quantum search algorithm
| x〉 ↦→ (−1) f(x) | x〉.
Let U be the following transformation in the n-qubit space:
U = 2 | 0〉〈0 | −I.
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