LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Normal form of a single-qubit ring unitary

Statement

If $U = (\begin{array}{cc} u & b \\ v & d \end{array})$ is a $2 \times 2$ unitary with entries in $D$ , then for some $j \in {0, \dots, 7}$ ,

$U = (\begin{array}{cc} u & - ω^{j} \overset{ˉ}{v} \\ v & ω^{j} \overset{ˉ}{u} \end{array}), ∣ u ∣^{2} + ∣ v ∣^{2} = 1.$

Facts & Assumptions

Given: $U^{*} U = I$ , and $u, v, b, d \in D$ .

[F1]

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces: in finite dimension a unitary operator satisfies $U^{*} U = U U^{*} = I$ .

[F2]

Unit-modulus elements of the dyadic cyclotomic ring: if $x \in D$ and $x \overset{ˉ}{x} = 1$ , then $x = ω^{j}$ for some integer $j$ .

Proof

1.1F1givenalgebra

The first column of $U$ is a unit vector, so $∣ u ∣^{2} + ∣ v ∣^{2} = 1$ . Its orthogonal complement is the one-dimensional span of $(- \overset{ˉ}{v}, \overset{ˉ}{u})$ , which is itself a unit vector. Therefore the second column is $λ (- \overset{ˉ}{v}, \overset{ˉ}{u})$ for a scalar $λ$ with $∣ λ ∣ = 1$ .

2.1F2step 1.1algebra∎

The determinant of this displayed matrix is $λ (∣ u ∣^{2} + ∣ v ∣^{2}) = λ$ . Because its entries lie in $D$ and $D$ is a ring, $λ = u d - v b \in D$ . By [F2], $λ = ω^{j}$ for some $j \in {0, \dots, 7}$ . Substitution gives the asserted form.

Unresolved supplier checks

The actual QC07 gate, QC09 circuit-semantics, and local entry-ring bodies were inspected; their matrix and ordered-word conventions match this argument. The supplier items remain draft and are not accepted by this record.

Depends on

Used by

Dependency tree · two levels

12 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

Kliuchnikov, Maslov, and Mosca, Practical approximation of single-qubit unitaries by single-qubit quantum circuits (standard reference, not scraped)