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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A prepared state with success amplitude one half

Example

Take an orthonormal pair $(∣ B ⟩, ∣ G ⟩)$ and the unit prepared state $∣ ψ ⟩ = \frac{\sqrt{3}}{2} ∣ B ⟩ + \frac{1}{2} ∣ G ⟩$ with $Π_{g o o d} = ∣ G ⟩ ⟨ G ∣$ . Its success amplitude is $1 / 2$ , so $a = 1 / 4$ and $θ = π / 6$ . Compute the good probabilities after $k = 0, 1, 2$ standard iterations.

Facts & Assumptions

Given: The state, projection, and iteration counts in the Example.

[F1]

Exact success probability after amplitude amplification: after $k$ iterations, $p_{k} = \sin^{2} ((2 k + 1) θ)$ .

[F2]

Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions gives $\sin (π / 6) = \cos (π / 3)$ and $\cos (2 π / 3) = - \cos (π / 3)$ ; Double-angle and quadratic power-reduction identities gives $\cos (2 x) = 2 \cos^{2} x - 1$ ; Pi is the first positive zero of sine gives $\sin x > 0$ for $0 < x < π$ .

[F3]

Quarter-turn values and shifts by pi/2 and pi gives $\sin (π / 2) = 1$ , and Cofunction, supplementary, quarter-turn, and reflection identities for the six trigonometric functions gives $\sin (5 π / 6) = \sin (π / 6)$ .

Verification

1.1givenF2algebra

Put $c = \cos (π / 3) = \sin (π / 6)$ by [F2]. Since $0 < π / 6 < π$ , [F2] gives $c > 0$ . Also [F2] gives $\cos (2 π / 3) = - c$ , while [F2] gives $\cos (2 π / 3) = 2 c^{2} - 1$ . Hence $2 c^{2} + c - 1 = (2 c - 1) (c + 1) = 0$ ; positivity excludes $c = - 1$ , so $\sin (π / 6) = c = 1 / 2$ . The vector is normalized because $3 / 4 + 1 / 4 = 1$ , and $∥ Π_{g o o d} ψ ∥^{2} = 1 / 4 = \sin^{2} (π / 6)$ .

2.1givenF1F3step 1.1algebra∎

By [F1], [F3], and step 1.1, $p_{0} = \sin^{2} (π / 6) = 1 / 4$ , $p_{1} = \sin^{2} (π / 2) = 1$ , and $p_{2} = \sin^{2} (5 π / 6) = 1 / 4$ . Each value lies in $[0, 1]$ ; the second iteration overshoots the exact peak attained after one iteration.

Depends on

Used by

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Sources

Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp, Quantum Amplitude Amplification and Estimation (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)