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Exact success probability after amplitude amplification

Statement

Let $a \in [0, 1]$ be the initial success weight and let $θ \in [0, π / 2]$ be the unique angle satisfying $\sin^{2} θ = a$ . After exactly $k \geq 0$ standard amplitude-amplification iterates, the probability of the good outcome of ${Π_{g o o d}, I - Π_{g o o d}}$ is $\sin^{2} ((2 k + 1) θ) .$ The endpoints $a = 0$ and $a = 1$ are included by direct evaluation, without normalizing a zero component.

Facts & Assumptions

Given: The sampler, success projection, and integer $k$ in the Statement.

[F1]

Amplitude-amplification invariant plane: for $0 < a < 1$ , the ordered $(∣ B ⟩, ∣ G ⟩)$ matrix of $Q$ is the rotation through $2 θ$ ; at $a = 0, 1$ , the prepared vector is fixed up to phase.

[F2]

A unit-vector outer product is a density operator: the outer product of a unit vector is a density operator.

[F3]

Projective measurement outcome weights are probabilities: for a density operator $ρ$ , the probability of the outcome projection $Π$ is $tr (ρ Π)$ .

[F4]

The addition formulas for complex trigonometric and hyperbolic functions: sine and cosine satisfy the real angle-addition identities; Parity and the Pythagorean identity for sine and cosine gives $\sin^{2} x + \cos^{2} x = 1$ .

[F5]

Quarter-turn values and shifts by pi/2 and pi: $\sin (π / 2) = 1$ and $\cos (π / 2) = 0$ .

[F6]

Signs, monotonicity intervals, and ranges of sine and cosine: sine is strictly increasing on $[0, π / 2]$ , so $\sin^{2} θ = a$ has a unique $θ \in [0, π / 2]$ for each $a \in [0, 1]$ .

Proof

1.1givenF1F4F6induction

Assume $0 < a < 1$ . The initial coordinates in the invariant ordered basis are $(\cos θ, \sin θ)$ . Multiplication by the matrix in [F1] and the addition formulas in [F4] send $(\cos t, \sin t)$ to $(\cos (t + 2 θ), \sin (t + 2 θ))$ . Starting at $t = θ$ , induction on $k$ therefore gives coordinates $(\cos ((2 k + 1) θ), \sin ((2 k + 1) θ))$ after $k$ iterates.

2.1givenF2F3step 1.1algebra

The good component is the second coordinate, so its squared norm is $\sin^{2} ((2 k + 1) θ)$ . The prepared unit vector gives a density operator by [F2], and [F3] identifies this squared component norm with the projective-measurement outcome probability.

3.1givenF1F5algebra∎

If $a = 0$ , then $θ = 0$ and [F1] gives success probability $0 = \sin^{2} ((2 k + 1) 0)$ . If $a = 1$ , then $θ = π / 2$ ; [F1] changes only the global sign on each iterate, and [F5] gives $\sin^{2} ((2 k + 1) π / 2) = 1$ . Thus the same formula holds at both endpoints.

Depends on

Used by

Dependency tree · two levels

55 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)