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.

Finite toric-code Hamiltonian

Definition

For $L \geq 3$ and couplings $J_{e}, J_{m} > 0$ , the finite toric-code Hamiltonian on $H_{L}$ is

$H_{L} = - J_{e} \sum_{i, j \in Z_{L}} A_{i, j} - J_{m} \sum_{i, j \in Z_{L}} B_{i, j} .$

The energy unit is fixed by $J_{e}, J_{m}$ . This is a finite-volume operator defined from the stabilizer checks; its definition makes no claim about a material realization or a thermodynamic phase.

Facts & Assumptions

Given: The finite register and commuting checks from Toric star and plaquette checks.

[F1]

Toric star and plaquette checks: every star and plaquette is a Hermitian involution supported on four edges.

[F2]

Toric checks commute and have two global relations: all checks commute.

Verification

1.1givenF1algebra

Each coefficient $- J_{e}$ and $- J_{m}$ is real, and each summand is self-adjoint by [F1]. Their finite sum is therefore self-adjoint.

2.1F1F2step 1.1∎

The terms commute pairwise by [F2], so $H_{L}$ is a finite sum of commuting local check operators. Its locality follows from the four-edge supports in [F1].

Depends on

Used by

A finite toric gap does not imply arbitrary perturbation stability Counterexample
Pauli-path energy barrier Definition
L=3 torus checks and syndrome calculation Example
A toric logical string has constant energy barrier Theorem
Exact finite-volume toric-code gap Theorem

Dependency tree · two levels

9 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)
John Preskill, Quantum Computation, Chapter 9: Topological Quantum Computation (standard reference, not scraped)