LemmaStatement: AI-adaptedProof: AI-generatedPipeline-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.

Toric checks commute and have two global relations

Statement

All star and plaquette checks of Toric star and plaquette checks commute. The product of all stars is the identity, and the product of all plaquettes is the identity.

Facts & Assumptions

Given: The periodic $L \times L$ torus with $L \geq 3$ and its star and plaquette checks.

[F1]

Toric star and plaquette checks: $A_{i, j}$ is $X$ on the four edges incident to $q_{i, j}$ , and $B_{i, j}$ is $Z$ on the four edges of $f_{i, j}$ .

[F2]

Pauli commutator is symplectic pairing: two Pauli operators commute when the mod-two symplectic pairing of their binary supports is zero; for an $X$ -support and a $Z$ -support, this pairing is their overlap parity.

[F3]

Periodic square cellulation of the torus: every edge has two distinct endpoints and borders two distinct faces, with all incidences read modulo $L$ .

Proof

1.1givenF1F2algebra

For a fixed vertex $q_{i, j}$ , the only faces sharing its incident edges are $f_{i, j}, f_{i - 1, j}, f_{i, j - 1}, f_{i - 1, j - 1}$ . Their boundaries overlap the star support respectively in ${h_{i, j}, v_{i, j}}$ , ${h_{i - 1, j}, v_{i, j}}$ , ${h_{i, j}, v_{i, j - 1}}$ , and ${h_{i - 1, j}, v_{i, j - 1}}$ ; every other face has empty overlap. Hence every star-plaquette overlap has even size. By [F2], each star commutes with each plaquette. Two stars commute because all their nonidentity factors are $X$ ; two plaquettes commute because all their nonidentity factors are $Z$ .

2.1F1F3algebra∎

By [F3], every edge has exactly two endpoints, so in the product of all stars its $X$ factor occurs twice. Since $X^{2} = I$ , the total product is $I$ . Every edge borders exactly two faces, so the same argument with $Z^{2} = I$ gives $\prod_{i, j} B_{i, j} = I$ . The first total product is also $\prod_{i, j} A_{i, j} = I$ .

Depends on

Used by

Finite toric-code Hamiltonian Definition
Noncontractible toric logical loops Definition
L=3 torus checks and syndrome calculation Example
Toric checks decompose into joint syndrome eigenspaces Lemma
Exact indistinguishability in a contractible disk Theorem
The periodic toric code encodes two qubits Theorem

Dependency tree · two levels

10 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

Alexei Kitaev, Fault-tolerant quantum computation by anyons (standard reference, not scraped)
Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, Topological quantum memory (standard reference, not scraped)