LemmaStatement: AI-adaptedProof: 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.

Normalized transport encoding is a finite simple graph of bounded size

Statement

Let $C$ be an $n$ -wire, $d$ -layer circuit and let $R$ be the number of rounds in the normalized schedule. Form a candidate vertex list $V_{T} = {v_{j, s} : 1 \leq j \leq n, 0 \leq s \leq R}$ , horizontal edges ${v_{j, s - 1}, v_{j, s}}$ , and vertical edges ${v_{j, s - 1}, v_{k, s - 1}}$ for each ${j, k} \in M_{s}$ . Each edge list is loop-free and duplicate-free, the lists are disjoint, and their union is a finite simple graph with exactly $n (R + 1)$ vertices and $R \leq 3 d$ ; thus the graph object used in the Definition is well-defined.

Facts & Assumptions

Given: A finite normalized transport schedule and the displayed vertex and edge-list construction.

[F1]

Every original layer contributes zero, one, two, or three rounds, each with a matching $M_{s}$ (Normalized transport schedule for a J/CZ circuit). The horizontal and vertical edge lists are defined in the Statement.

[F2]

A finite simple graph has a finite vertex set and an edge set whose members are two-element subsets of the vertex set (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets).

Proof

1.1F1algebracases

Each original layer contributes at most three rounds by [F1], hence $0 \leq R \leq 3 d$ . The vertex list is the Cartesian product of $n$ wire labels and $R + 1$ column labels, so it has exactly $n (R + 1)$ vertices. If $n = 0$ , all layers are empty and $R = 0$ ; if $n = 1$ , every matching is empty.

1.2F1algebra

A horizontal edge has endpoints in distinct consecutive columns. Two horizontal edges can be the same unordered pair only if their wire and consecutive column pair agree, so the horizontal list has no loops or duplicates. A vertical edge has two distinct wire endpoints because each matching edge joins distinct wires, and no unordered wire pair repeats in a matching. These are the raw lists specified in the Statement.

2.1F1step 1.2algebra

Vertical edges from different rounds lie in different columns. A vertical edge has both endpoints in one column, while a horizontal edge has endpoints in two columns; thus the lists are disjoint. Together with step 1.2, the edge list is a finite list of distinct two-element subsets of the finite vertex list.

3.1step 1.1step 1.2step 2.1F2∎

The vertex set and edge set from step 1.1 and step 1.2 and step 2.1 therefore satisfy the finite-simple-graph definition [F2].

Depends on

Used by

Dependency tree · two levels

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Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415, §20.6 (standard reference, not scraped)
Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, Measurement-based quantum computation on cluster states, §§II D–F (standard reference, not scraped)