t b a b a

|Good>
θ
| ψ >
|ψ 0 >
|Bad>
|Good>
θ
| ψ >
|ψ 0 >
|Bad>
|Good>
(a) (b) (c)
2θ
θ
| ψ >
|ψ 0 >
|Bad>
Figure 3.3: The search operator Q performs a rotation toward | Good〉 states by 2θ radians.
(a) The current state | ψ〉 and the initial state | ψ0〉. (b) The oracle operation Sχ performs a
reflection of the current state | ψ〉 about the vector | Good〉. (c) AS0A −1 performs a reflection
about the initial state | ψ0〉
state consisting of a uniform superposition of solutions to the search problem. The initial
state may be expressed as:
| ψ0〉 = √ a | Good〉 + √ 1 − a | Bad〉
Figure 3.3 presents the effect of the transformation Q = −AS0A −1 Sχ as:
• the oracle operation Sχ performs a reflection about the vector | Good〉.
Sχ | x〉 =| x〉 (χ(x) = 1) Sχ | x〉 = − | x〉 (χ(x) = 0)
• AS0A −1 performs a reflection about the initial state | ψ0〉
S0 | 0〉 =| 0〉 S0 | x〉 = − | x〉 (x �= 0)
• Q = −AS0A −1 Sχ performs a rotation toward | Good〉 vector by 2θ radians, where
sin 2 θ = a
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