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Success guarantee and boundary behavior of known-count Grover search

Statement

For exact known count $t > 0$ , the procedure Known-count Grover search by exact integer maximization outputs a marked item with probability at least $\max {t / N, 1 - t / N}$ . It returns none with certainty when $t = 0$ and returns a marked item with certainty when $t = N$ .

Facts & Assumptions

Given: $N = 2^{n}$ , a marked set $M$ , and exact count $t = ∣ M ∣$ supplied to the algorithm.

[F1]

Known-count Grover search by exact integer maximization: if $t = 0$ it returns none; otherwise it chooses the least maximizer of the rational values $r_{k} = (1 - T_{2 k + 1} (q)) / 2$ over $0 \leq k \leq K = ⌈ \sqrt{N / t} ⌉$ , where $q = 1 - 2 t / N$ . It applies that many Grover iterates, measures an address, and verifies it with the exact bit oracle for $f = 1_{M}$ .

[F2]

Exact success probability after an integer number of Grover iterates: for $0 < t < N$ and $θ = \arcsin \sqrt{t / N}$ , after $k$ iterates the marked probability is $p_{k} = \sin^{2} ((2 k + 1) θ)$ . For $t = 0, N$ it is respectively $0, 1$ for every integer $k \geq 0$ .

[F3]

Integer rounding gives a high-success Grover iterate: for $0 < t / N \leq 1 / 2$ , an integer $k \leq K$ has $p_{k} \geq 1 - t / N$ ; for $t / N > 1 / 2$ , $k = 0$ has $p_{0} = t / N$ .

[F4]

Bit cost of exact known-count iteration selection: exact selection computes $T_{0} = 1$ , $T_{1} = q$ , $T_{m + 1} = 2 q T_{m} - T_{m - 1}$ through $2 K + 1$ and compares the finite rational list $(1 - T_{2 k + 1} (q)) / 2$ .

[F5]

Double-angle and quadratic power-reduction identities: $\cos (2 u) = 1 - 2 \sin^{2} u$ and $\sin^{2} u = (1 - \cos (2 u)) / 2$ .

[F6]

The addition formulas for sine and cosine: $\cos (u + v) = \cos u \cos v - \sin u \sin v$ .

[F7]

Parity and the Pythagorean identity for sine and cosine: cosine is even and sine is odd.

[F8]

The derivatives of sine and cosine are cosine and minus sine: $\cos 0 = 1$ .

[F9]

Principal inverse sine and inverse cosine: $\sin (\arcsin y) = y$ for $y \in [- 1, 1]$ .

[F10]

Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ : $\sqrt{a} \geq 0$ and $(\sqrt{a})^{2} = a$ for $a \geq 0$ .

Proof

1.1givenF1F2

If $t = 0$ , [F1] returns none. If $t = N$ , [F2] gives marked probability one for every integer iteration count, and the exact verification in [F1] accepts every address. Thus the two endpoint claims hold.

1.2givenF1F2F4F5F6F7F8F9F10inductionalgebra

For $0 < t < N$ , put $a = t / N$ and $θ = \arcsin \sqrt{a}$ . By [F9]–[F10], $\sin^{2} θ = a$ , so [F5] gives $q = 1 - 2 a = \cos (2 θ)$ . The recurrence in [F4] starts with $T_{0} = \cos 0 = 1$ and $T_{1} = \cos (2 θ)$ by [F8]. Expanding cosine at $2 m θ \pm 2 θ$ using [F6]–[F7] gives $\cos (2 (m + 1) θ) + \cos (2 (m - 1) θ) = 2 \cos (2 θ) \cos (2 m θ)$ . Induction in that recurrence therefore gives $T_{m} (q) = \cos (2 m θ)$ at every computed index. Consequently $r_{k} = (1 - \cos ((4 k + 2) θ)) / 2 = \sin^{2} ((2 k + 1) θ) = p_{k}$ for every candidate $0 \leq k \leq K$ , by [F5] and [F2].

2.1givenF1F3step 1.2algebra

If $0 < a \leq 1 / 2$ , the witness $k \leq K$ from [F3] belongs to the algorithm's finite candidate list and has $p_{k} \geq 1 - a$ . Step 1.2 identifies each rational candidate value with its actual marked probability, so the selected maximizer has success at least $1 - a = \max {a, 1 - a}$ . Exact verification accepts every marked measurement by [F1].

3.1givenF1F3step 1.1step 1.2algebra∎

If $1 / 2 < a < 1$ , the candidate $k = 0$ in [F3] has success $a > 1 / 2$ . Step 1.2 identifies the maximized list with actual marked probabilities, so the selected iterate succeeds with probability at least $a = \max {a, 1 - a}$ . Exact verification again accepts every marked measurement by [F1]. Together with step 1.1, this proves every branch of the guarantee.

Depends on

Used by

Known-count Grover search uses O of the square root of N over t bit queries Theorem

Dependency tree · two levels

54 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

Ronald de Wolf, Quantum Computing: Lecture Notes, §§7.1–7.2 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, §19.1 (standard reference, not scraped)
Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp, Quantum Amplitude Amplification and Estimation, §2 (standard reference, not scraped)