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

Graph States and Measurement-Based Computation — Examples

1 · Prerequisites

2 · Summary

The one-edge graph gives a two-qubit state with one negative computational-basis amplitude and generators $X \otimes Z$ and $Z \otimes X$ . A four-vertex open-graph pattern separately checks all four outcomes of the CZ gadget, including its branch probability and exact Pauli/Hadamard correction convention.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

One-edge graph state and two-wire CZ pattern

Example

For the graph with vertices $1, 2$ and one edge ${1, 2}$ ,

$∣ G ⟩ = C Z ∣ + + ⟩ = \frac{1}{2} (∣ 00 ⟩ + ∣ 01 ⟩ + ∣ 10 ⟩ - ∣ 11 ⟩) .$

Its two graph-state generators are $K_{1} = X \otimes Z$ and $K_{2} = Z \otimes X$ . Separately, the four-vertex pattern with inputs $1, 2$ , outputs $3, 4$ , and edges ${1, 2}, {1, 3}, {2, 4}$ has branch $\frac{1}{2} (X^{b_{1}} H \otimes X^{b_{2}} H) C Z$ and terminal correction $H X^{b_{1}} \otimes H X^{b_{2}}$ . In particular, on input $∣ 00 ⟩$ all four uncorrected outputs are $\frac{1}{2} ∣ + + ⟩$ , each with probability $1 / 4$ , and all corrected outputs are $\frac{1}{2} ∣ 00 ⟩$ .

Facts & Assumptions

Given: The one-edge simple graph and the specified four-vertex open graph pattern.

[F1]

Graph-state amplitudes are $2^{- n / 2}$ times the edge-parity sign (Computational-basis amplitudes of a graph state).

[F2]

For each vertex, $K_{v} = X_{v} \prod_{u \in N (v)} Z_{u}$ fixes the graph state (Graph-state stabilizer generators).

[F3]

The four-vertex CZ pattern has the stated branch and terminal-correction formula (Four-branch open-graph pattern for CZ).

Verification

1.1F1algebra

In the one-edge graph, the edge-parity exponent is $x_{1} x_{2}$ , so the four ordered bitstrings $00, 01, 10, 11$ have signs $+ 1, + 1, + 1, - 1$ in [F1]. This gives the displayed normalized vector.

1.2F1F2algebra

The neighbors are $N (1) = {2}$ and $N (2) = {1}$ , so [F2] gives $K_{1} = X \otimes Z$ and $K_{2} = Z \otimes X$ . Directly, $X \otimes Z$ sends the four basis vectors to $∣ 10 ⟩, - ∣ 11 ⟩, ∣ 00 ⟩, - ∣ 01 ⟩$ , respectively; applied to the displayed coefficient vector this reproduces it. Likewise $Z \otimes X$ sends them to $∣ 01 ⟩, ∣ 00 ⟩, - ∣ 11 ⟩, - ∣ 10 ⟩$ , which also reproduces the vector.

2.1F3algebra∎

For input $∣ 00 ⟩$ , $C Z ∣ 00 ⟩ = ∣ 00 ⟩$ and $H ∣ 0 ⟩ = ∣ + ⟩$ . Thus [F3] gives each branch output $\frac{1}{2} ∣ + + ⟩$ , whose squared norm is $1 / 4$ . Since $H X^{b_{j}} X^{b_{j}} H ∣ 0 ⟩ = ∣ 0 ⟩$ for $b_{j} \in {0, 1}$ , each terminal correction gives $\frac{1}{2} ∣ 00 ⟩$ .