ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Surface-Code Memory Bound Under IID Z Noise

Statement

Assume the distance-five relative square planar patch, which has $n = 2 (5)^{2} = 50$ data qubits, iid $Z$ -error rate $ϵ = 1 / 144$ , one perfect syndrome round, and minimum-weight recovery. The finite series in the planar memory theorem gives the conditional upper bound $P_{l o g i c a l} \leq 6 \sum_{ℓ = 5}^{50} (1 / 2)^{ℓ} = \frac{3}{8} (1 - 2^{- 46}) < \frac{3}{8}$ . This calculation concerns an ideal-syndrome memory round only.

Example

Take the stated finite parameters. Since $36 ϵ = 1 / 4$ , each summand $(36 ϵ)^{ℓ / 2}$ equals $2^{- ℓ}$ . The theorem's prefactor is $d + 1 = 6$ .

Facts & Assumptions

Given: $d = 5$ , $n = 50$ , $ϵ = 1 / 144$ , iid $Z$ -noise, perfect syndrome extraction, and minimum-weight recovery.

[F1]

Planar Surface-Code Memory Bound Under IID Z Noise: Under those hypotheses, $P_{l o g i c a l} \leq (d + 1) \sum_{ℓ = d}^{n} (36 ϵ)^{ℓ / 2}$ .

Verification

1.1givenalgebra

Compute $36 ϵ = 36 / 144 = 1 / 4$ , so $(36 ϵ)^{ℓ / 2} = ((1 / 4)^{1 / 2})^{ℓ} = 2^{- ℓ}$ .

2.1givenstep 1.1algebra

The finite geometric sum is $\sum_{ℓ = 5}^{50} 2^{- ℓ} = 2^{- 5} (1 - 2^{- 46}) / (1 - 1 / 2) = \frac{1}{16} (1 - 2^{- 46})$ . Multiplying by $d + 1 = 6$ gives $\frac{3}{8} (1 - 2^{- 46}) < \frac{3}{8}$ .

3.1F1step 2.1∎

Applying [F1] yields the stated conditional failure-probability bound. The hypotheses include perfect syndrome extraction; this example makes no faulty-gate threshold claim.

Depends on

Planar Surface-Code Memory Bound Under IID Z Noise

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

John Preskill, Quantum Computation, Chapter 7: Quantum Error Correction (standard reference, not scraped)