DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

J gate and ordered equatorial measurement vectors

Definition

For $α \in R$ , let $P_{α} = diag (1, e^{i α})$ and $J (α) = H P_{α}$ , with $H$ and the exact one-qubit gate convention from Exact one-qubit gates and rotations. Define the ordered measurement vectors

$∣ b ⟩_{α} : = J (α)^{†} ∣ b ⟩, b \in {0, 1},$

where $∣ 0 ⟩, ∣ 1 ⟩$ are the ordered computational basis vectors. The adjoint and complex inner product use the root convention translated by Dirac Notation Conversion. The global phase in $J (α)$ is fixed by the displayed matrix product; it is not suppressed.

Depends on

Used by

Circuit-to-open-graph resource encoding with normalized transport rounds Definition
Finite open-graph adaptive measurement pattern Definition
Normalized transport schedule for a J/CZ circuit Definition
Adaptive J angle and Pauli-frame update Lemma
J-gate basis and exact phase identities Lemma
One-bit J-gate teleportation branch identity Lemma
Circuit and pattern translations are branch preserving and universal Theorem
Pattern branch probabilities and zero-probability transcripts Theorem

Dependency tree · two levels

17 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, arXiv:1907.09415, §20.6 (standard reference, not scraped)
Robert Raussendorf and Hans J. Briegel, Quantum computing via measurements only, arXiv:quant-ph/0010033 (standard reference, not scraped)