ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generated

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.

Exact T period and a non-exact Rz phase

Example

Let $T = diag (1, ω)$ with $ω = e^{i π / 4}$ , and let $ζ_{64} = e^{2 π i / 64}$ . The gate $T$ has exact period eight, while $R_{z} (π / 16)$ is not an exact one-qubit Clifford+T word.

Facts & Assumptions

Given: $ω = e^{i π / 4}$ , $ζ_{64} = e^{i π / 32}$ , and $R_{z} (θ) = diag (e^{- i θ / 2}, e^{i θ / 2})$ .

[F1]

Clifford+T circuit entries lie in the dyadic cyclotomic ring: every exact one-qubit Clifford+T word has entries in $D = Z [ζ_{8}, 1 / 2] \subseteq Q (ζ_{8})$ .

[F2]

$Φ_{n}$ is irreducible in $Q [t]$ for every $n \geq 1$ : the cyclotomic polynomial $Φ_{n}$ is irreducible over $Q$ .

[F3]

The recursion defines a unique monic $Φ_{n} \in Z [t]$ , of degree $φ (n)$ : $\deg Φ_{n} = φ (n)$ .

Verification

1.1givenalgebra

Since $ω^{8} = 1$ , direct matrix powering gives $T^{8} = diag (1, ω^{8}) = I$ . No smaller positive power is the identity because $ω$ has order eight.

2.1F1F2F3givenalgebra∎

The lower-right entry of $R_{z} (π / 16)$ is $e^{i π / 32} = ζ_{64}$ . If an exact H/T word equaled this rotation, [F1] would put $ζ_{64}$ in $Q (ζ_{8})$ . But $ζ_{8} = ζ_{64}^{8}$ , so $Q (ζ_{8}) \subseteq Q (ζ_{64})$ ; containing $ζ_{64}$ in the smaller field would force equality. By [F2]–[F3], these fields have degrees $φ (8) = 4$ and $φ (64) = 32$ respectively: among residues modulo $2^{m}$ , exactly the $2^{m - 1}$ odd residues are coprime. The unequal degrees contradict equality, so no exact H/T word implements this rotation.

Remarks

The example uses the necessary direction of the ring characterization only; it does not rely on the converse theorem or on an approximation algorithm.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

40 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, Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates (standard reference, not scraped)
de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)