DefinitionDefinition: 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 star and plaquette checks

Definition

For the periodic cellulation in Periodic square cellulation of the torus, put one qubit on each edge, ordered lexicographically by $(i, j)$ and then with horizontal before vertical edges. The physical Hilbert space is $H_{L} = (C^{2})^{\otimes E}$ , where $E$ is the edge set. For an edge $e$ , write $X_{e}$ and $Z_{e}$ for the corresponding one-qubit Pauli acting on that tensor factor and as the identity on every other factor. Define the star and plaquette checks by

$A_{i, j} = X_{h_{i - 1, j}} X_{h_{i, j}} X_{v_{i, j - 1}} X_{v_{i, j}}, B_{i, j} = Z_{h_{i, j}} Z_{v_{i + 1, j}} Z_{h_{i, j + 1}} Z_{v_{i, j}} .$

All indices are in $Z_{L}$ .

Facts & Assumptions

Given: An integer $L \geq 3$ , its periodic square cellulation, and the ordered edge register.

[F1]

Periodic square cellulation of the torus: the edges incident to $q_{i, j}$ are $h_{i - 1, j}, h_{i, j}, v_{i, j - 1}, v_{i, j}$ ; the boundary of $f_{i, j}$ is $h_{i, j}, v_{i + 1, j}, h_{i, j + 1}, v_{i, j}$ , with four distinct edges in each list.

[F2]

The Pauli matrices: $X$ and $Z$ are the specified Hermitian one-qubit matrices, and $X^{2} = Z^{2} = I$ by direct multiplication.

[F3]

Finite registers and their ordered computational bases: the physical space is the ordered tensor product of one copy of $C^{2}$ for each edge.

[F4]

Tensor products of finitely many linear maps: tensor products of maps act on elementary tensors factor by factor and extend linearly.

[F5]

The product basis is orthonormal and tensor dimensions multiply: ordered elementary computational-basis tensors span the finite register.

Verification

1.1givenF1F2F3

By [F1], each displayed check acts on four distinct edge factors. By [F2] and [F3], its factors are defined endomorphisms of the ordered physical register, so both $A_{i, j}$ and $B_{i, j}$ are well-defined tensor-product operators.

2.1F2F3F4F5step 1.1algebra∎

Each factor in either product is Hermitian and squares to the identity by [F2]. The factors act on distinct tensor coordinates, so [F4] makes their two compositions agree on every elementary tensor; those tensors span the register by [F5]. Therefore each full product is Hermitian and squares to the identity. These are the star and plaquette checks used below.

Depends on

Used by

Finite toric-code Hamiltonian Definition
Noncontractible toric logical loops Definition
Toric e and m syndromes Definition
L=3 torus checks and syndrome calculation Example
Open toric strings have endpoint syndromes Lemma
Toric checks commute and have two global relations Lemma
Toric checks decompose into joint syndrome eigenspaces Lemma
Exact indistinguishability in a contractible disk Theorem
Toric code distance is L Theorem

Dependency tree · two levels

18 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)