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Finite abelian hidden-subgroup reconstruction succeeds

Statement

Let $G$ be finite abelian, let $H \leq G$ be hidden by the supplied function, and let $0 < δ < 1$ . The common-kernel list output by Finite abelian hidden-subgroup common-kernel reconstruction always denotes a subgroup $K$ with $H \leq K$ . For $∣ G ∣ = 1$ it equals $H = G$ exactly without queries. For $∣ G ∣ \geq 2$ , set $s = ⌈ \log_{2} ∣ G ∣ ⌉$ and pass the integer $M = ⌈ 4 s + 16 \ln (1 / δ) ⌉$ to the reconstruction algorithm; its output equals $H$ with probability at least $1 - δ$ .

Facts & Assumptions

Given: The exact Fourier-sample procedure and the common-kernel reconstruction procedure with the integer $M$ specified in the Statement.

[F1]

Every sample lies in $H^{⊥}$ and the sample law is uniform there. (Fourier samples are uniform in the annihilator)

[F2]

If $K > H$ , a fresh sample cuts $K$ by index at least two with conditional probability at least $1 / 2$ , while preserving $H$ . (A Fourier sample cuts a larger common kernel)

[F3]

After $M = ⌈ 4 s + 16 \ln (1 / δ) ⌉$ independent samples, the probability that the final common kernel is not $H$ is at most $δ$ . (Chernoff repetition bound for common-kernel recovery)

Proof

1.1givenconstruct

If $∣ G ∣ = 1$ , its only subgroup is $G$ , and the algorithm outputs the empty constraint list denoting $G$ . Thus the conclusion holds with probability one and zero queries.

1.2F1given

Suppose $∣ G ∣ \geq 2$ . Each measured character belongs to $H^{⊥}$ by [F1], so it is trivial on $H$ . Hence every kernel in the output list contains $H$ , and their intersection $K$ satisfies $H \leq K$ .

1.3F2givenalgebra

Starting with $K_{0} = G$ , whenever $K_{t - 1} > H$ the conditional cut probability and index reduction are those of [F2]. After $s = ⌈ \log_{2} ∣ G ∣ ⌉$ cuts, the subgroup order is at most $∣ G ∣ / 2^{s} \leq 1$ ; since it contains the nonempty subgroup $H$ , it must equal $H$ .

2.1F3step 1.2step 1.3∎

The reconstruction procedure makes exactly $M$ independent samples. Therefore [F3] bounds the event that the final common kernel is not $H$ by $δ$ . Combining with $H \leq K$ from step 1.2 gives success probability at least $1 - δ$ .

Remarks

No subgroup generator extraction is used. The confidence integer is supplied to the reconstruction procedure; the theorem does not hide its classical computation.

Depends on

Used by

Finite abelian hidden-subgroup query and circuit costs Theorem

Dependency tree · two levels

45 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

Andrew M. Childs, Lecture Notes on Quantum Algorithms (17 April 2025) (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (standard reference, not scraped)
Andrew Aspnes, Notes on Randomized Algorithms, §§5.2.1–5.2.4 and H.2.2 (standard reference, not scraped)